MCAT Physics and Math Review

Chapter 3: Thermodynamics

3.3 First Law of Thermodynamics

We have already encountered the first law of thermodynamics in our discussion of the conservation of mechanical energy in Chapter 2 of MCAT Physics and Math Review. Remember that in the absence of nonconservative forces, the sum of kinetic and potential energies is constant in a system. Now, in our present discussion of thermodynamics, we will look more closely at the relationship between internal energy, heat, and work. Essentially, the first law of thermodynamics states that the change in the total internal energy of a system is equal to the amount of energy transferred in the form of heat to the system, minus the amount of energy transferred from the system in the form of work. The internal energy of a system can be increased by adding heat, doing work on the system, or some combination of both processes. The change in internal energy is calculated from the equation

ΔU = Q − W

Equation 3.4

where ΔU is the change in the system’s internal energy, Q is the energy transferred into the system as heat, and W is the work done by the system. To use this equation properly, one must carefully apply the following sign convention shown in Table 3.2.

Variable

Positive Value

Negative Value

Change in Internal Energy (ΔU)

Increasing temperature

Decreasing temperature

Heat (Q)

Heat flows into system

Heat flows out of system

Work (W)

Work is done by the system (expansion)

Work is done on the system (compression)

Table 3.2. Sign Convention for the First Law of Thermodynamics

KEY CONCEPT

The first law of thermodynamics tells us that an increase in the total internal energy of a system is caused by transferring heat into the system or performing work on the system. The total internal energy of a system will decrease when heat is lost from the system or work is performed by the system.

The first law is really just a particular iteration of the more universal physical law of energy conservation: energy can be neither created nor destroyed; it can only be changed from one form to another. Because the first law accounts for all work and all heat processes impacting the system, the presence of nonconservative forces poses no problem because the energy transfer associated with friction, air resistance, or viscous drag will be accounted for in the first law equation. For example, when a car “burns rubber,” all the smoke and noise coming from the back tires is a clear indication that mechanical energy is not being conserved. However, if we include the energy transfers associated with the frictional forces in our consideration of the change in internal energy of the system, then we can confidently say that no energy has been lost at all: there may be a “loss” of energy from the car as a result of the friction, but that precise amount of energy can be “found” elsewhere—as thermal energy in the atoms and molecules of the surrounding road and air.

HEAT

In Chapter 2 of MCAT Physics and Math Review, we defined work as the process by which energy is transferred as the result of force being applied through some distance. We noted that work and heat are the only two processes by which energy can be transferred from one object to another. As discussed earlier in this chapter, the zeroth law of thermodynamics says that objects in thermal contact are in thermal equilibrium when their temperatures are the same. The corollary of this is the second law of thermodynamics: objects in thermal contact and not in thermal equilibrium will exchange heat energy such that the object with a higher temperature will give off heat energy to the object with a lower temperature until both objects have the same temperature at thermal equilibrium. Heat, then, is defined as the process by which a quantity of energy is transferred between two objects as a result of a difference in temperature. As we will discuss further in our examination of the second law, heat can never spontaneously transfer energy from a cooler object to a warmer one without work being done on the system.

KEY CONCEPT

Heat is the process of energy transfer between two objects at different temperatures and will continue until the two objects come into thermal equilibrium at the same temperature.

The SI unit for heat is the joule (J), which should not be surprising because it is based on energy. Heat can also be measured in the units of calorie (cal), nutritional Calorie (Cal), or the British thermal unit (BTU). The nutritional Calorie (“big C”) is not the same thing as the calorie (“little c”); one Calorie is equal to 1000 calories or 1 kcal.

The conversion factors between the units of heat are as follows:

1 Cal ≡ 103 cal = 4184 J = 3.97 BTU

KEY CONCEPT

One calorie (little c) is the amount of heat required to raise 1 g of water one degree Celsius. One Calorie (big C) is the amount of heat required to raise 1 kg of water 1 degree Celsius, equal to 1000 calories.

Heat Transfer

For energy to be transferred between objects, they must be in thermal contact with each other. This does not necessarily mean that the objects are touching. Like force, energy can travel tremendous distances and does not require a medium to pass through. There are three means by which heat can transfer energy: conduction, convection, and radiation.

Conduction is the direct transfer of energy from molecule to molecule through molecular collisions. As this definition would suggest, there must be direct physical contact between the objects. At the atomic level, the particles of the hotter matter transfer some of their kinetic energy to the particles of the cooler matter through collisions between the particles of the two materials. Metals are described as the best heat conductors because metallic bonds contain a density of atoms embedded in a sea of electrons, which facilitate rapid energy transfer. Gases tend to be the poorest heat conductors because there is so much space between individual molecules that energy-transferring collisions occur relatively infrequently. An example of heat transfer through conduction is the heat that is rapidly, and painfully, conducted to your fingers when you touch a hot stove.

Convection is the transfer of heat by the physical motion of a fluid over a material. Because convection involves flow, only liquids and gases can transfer heat by this means. In convection, if the fluid has a higher temperature, it will transfer energy to the material. Most restaurants and some home kitchens have convection ovens, which use fans to circulate hot air inside the oven. Because the heat is being transferred to the food by both convection and radiation rather than only by radiation, convection ovens cook more rapidly than radiation-only ovens. Convection may also be used to wick heat away energy from a hot object. In laboratory experiments, for example, a running cold water bath may be used to rapidly cool a reaction.

Radiation is the transfer of energy by electromagnetic waves. Unlike conduction and convection, radiation can transfer energy through a vacuum. Radiation is the method by which the Sun is able to warm the Earth. Most home kitchens have radiant ovens, which use either electrical coils or gas flames to heat the insulated metal box that forms the body of the oven. The hot metal box then radiates the energy through the open space of the oven, where it is absorbed by whatever food is placed inside.

Specific Heat

When heat energy is added to or removed from a system, the temperature of that system will change in proportion to the amount of heat transfer, unless the system is undergoing a phase change during which the temperature is constant. This relationship between heat and temperature for a substance is called the specific heat. The specific heat (c) of a substance is defined as the amount of heat energy required to raise one gram of a substance by one degree Celsius or one unit kelvin. For example, the specific heat of liquid water is one calorie per gram per unit kelvin Equivalently, this can be expressed as  The specific heat for a substance changes according to its phase. The MCAT will generally provide specific heat values as necessary, although you are expected to know the specific heat of water in calories. The equation that relates the heat gained or lost by an object and the change in temperature of that object is

q = mcΔT

Equation 3.5

where m is the mass, c is the specific heat of the substance, and ΔT is the change in temperature (in Celsius or kelvin). Because the unit size for the Celsius and Kelvin scales is the same, the change in temperature will be the same for temperatures measured in Celsius or kelvin.

MCAT EXPERTISE

The specific heat of water (in calories) is a constant you are expected to know for Test Day. Its value is 

MNEMONIC

The equation for heat transfer, given a specific heat, is the same as the test you’re studying for! q = mcΔT looks a lot like “q equals MCAT.”

Heat of Transformation

When a substance is undergoing a phase change, such as from solid to liquid or liquid to gas, the heat that is added or removed from the system does not result in a change in temperature. In other words, phase changes occur at a constant temperature, and the temperature will not begin to change until all of the substance has been converted from one phase into the other. For example, water melts at 0°C. No matter how much heat is added to a mass of ice at 0°C, the temperature of the equilibrated system will not rise until all the ice has been melted into liquid water.

We’ve determined that adding heat raises the temperature of a system because the particles in that system now have a greater average kinetic energy, and it’s true that molecules have greater degrees of freedom of movement in the liquid state than in the solid state (and even more so in the gas state). However, phase changes are related to changes in potential energy, not kinetic energy. The molecules of water in ice, for example, aren’t truly frozen in place and unable to move. The molecules rotate, vibrate, and wiggle around. The bonds within each molecule are also free to bend and stretch. Of course, the molecules are held in relatively stable positions by the hydrogen bonds that form between them, but they still have a fairly significant amount of kinetic energy. The potential energy, however, is quite low because of the stability provided by the relative closeness of one molecule to another and by the hydrogen bonds.

Now, think about what happens when one adds heat to ice that is at 0°C. The heat energy causes the water molecules to begin to move away from each other by breaking free of the hydrogen bonds between them. Because the water molecules are being held less rigidly in place, they now have greater degrees of freedom of movement and their average potential energy increases. In statistical mechanics, one would say that this increased freedom of movement permits a greater number of microstates for the water molecules. For example, instead of only being able to move up and down or sway side-to-side, a water molecule may now have more freedom of movement and be able to rock forward and back. In gaining additional directions and forms of motion, however, the amount of up-and-down or side-to-side motion must decrease, thus keeping the average kinetic energy of liquid water at 0°C the same as solid water at 0°C. In summary, while liquid water may have a greater number of microstates due to increased freedom of movement, its average kinetic energy is the same as solid water at the same temperature.

When heat energy is added to or removed from a system that is experiencing a phase change, the amount of heat that is added or removed cannot be calculated with the equation q = mcΔT because there is no temperature change during a phase change. Instead, the following equation is used:

q = mL

Equation 3.6

where q is the amount of heat gained or lost from the material, m is the mass of the substance, and L is the heat of transformation or latent heat of the substance.

BRIDGE

It is important to know the common terms used for phase changes:

·        Solid to liquid: fusion or melting

·        Liquid to solid: freezing or solidification

·        Liquid to gas: boiling, evaporation, or vaporization

·        Gas to liquid: condensation

·        Solid to gas: sublimation

·        Gas to solid: deposition

These phase changes are discussed in Chapter 7 of MCAT General Chemistry Review.

The phase change from liquid to solid (freezing or solidification) or solid to liquid (melting or fusion) occurs at the melting point. The corresponding heat of transformation is called the heat of fusion. The phase change from liquid to gas (boilingevaporation, or vaporization) or gas to liquid (condensation) occurs at the boiling point. The corresponding heat of transformation is called the heat of vaporization. The relevant heats of fusion and vaporization will be provided on Test Day.

REAL WORLD

It is because of the heat of transformation that sweating is such an efficient cooling mechanism. When sweat evaporates, the heat of vaporization is absorbed from the surface of the body. This is also why a hot day seems so much more intense when it is very humid out. The sweat is less likely to evaporate due to the dampness of the environment, so less heat can be lost from the surface of the skin through sweating.

Example:

Silver has a melting point of 962°C and a heat of fusion of approximately  The specific heat of silver is  Approximately how much heat is required to completely melt a 1 kg silver chain with an initial temperature of 20°C?

Solution:

Before melting the chain, we must first heat the chain to the melting point. To figure out how much heat is required, we use this formula:

This tells us we have to add 219 kJ of heat to the chain just to get its temperature to the melting point. The chain is still in the solid phase. To melt it, we must continue to add heat in accordance with this formula:

The total heat needed to melt the solid silver chain is 219 kJ + 105 kJ = 324 kJ.

THERMODYNAMIC PROCESSES

In the last chapter, we gave significant consideration to work as a change of energy in a system, both as a function of force and displacement and as a function of volume and pressure. We will briefly review the latter and its relationship to heat transfer within a system. Keep in mind that work accomplished by a change in displacement is not likely to be motivated by heat transfer, and any heat transfer that does occur is most likely a result of friction dissipating mechanical energy from the system.

During any thermodynamic process, a system goes from some initial equilibrium state with an initial pressure, temperature, and volume to some other equilibrium state, which may be at a different final pressure, temperature, or volume. These thermodynamic processes can be represented in graphical form with volume on the x-axis and pressure (or temperature) on the y-axis.

Process

First law of thermodynamics reduces to:

Isothermal (ΔU = 0)

Q = W

Adiabatic (Q = 0)

ΔU = −W

Isobaric (constant pressure)

(No special form)

Isovolumetric (isochoric) (W = 0)

ΔU = Q

Table 3.3. Special Types of Thermodynamic Processes

The MCAT focuses on three particular thermodynamic processes as special cases of the first law, as shown in Table 3.3. In each of these cases, some physical property is held constant during the process. These processes are isothermal (constant temperature, and therefore no change in internal energy), adiabatic (no heat exchange), and isovolumetric (no change in volume, and therefore no work accomplished; also called isochoric). Isobaric processes are those that occur at a constant pressure. Figure 3.1 shows the different types of thermodynamic behaviors for a gas.

Figure 3.1. Thermodynamic Behaviors of Gases

Figure 3.2 shows a closed-loop thermodynamic process. Because the work on a P–V graph is simply the area under the curve, the work done in this closed-loop process is the area inside the loop.

Figure 3.2. A Closed-Loop Process The work is the area inside the curve.

Example:

A gas in a cylinder is kept at a constant pressure of 3.5 × 105 Pa while 300 kJ of heat are added to it, causing the gas to expand from 0.9 m3 to 1.5 m3. Find the work done by the gas and the change in internal energy of the gas.

Solution:

The pressure is held constant through the entire process so the work can be found using the equation:

The change in internal energy can be found from the first law of thermodynamics:

ΔU = QW = 3 × 105 J − 2.1 × 105 J = 0.9 × 105 J = 90 kJ

MCAT Concept Check 3.3:

Before you move on, assess your understanding of the material with these questions.

1.    Describe the relationship between internal energy, work, and heat in words.

2.    Define the following forms of heat transfer:

·        Conduction:

·        Convection:

·        Radiation:

3.    Draw a representative graph of the temperature of a solid object as it is heated and goes through two phase changes to become a gas.

4.    How is work calculated in P–V diagrams?