MCAT Physics and Math Review

Chapter 3: Thermodynamics

Introduction

Thermodynamics is the study of the flow of energy in the universe, as that flow relates to work, heat, entropy, and the different forms of energy. Classical thermodynamics concerns itself only with observations that can be made at the macroscopic level, such as measurements of temperature, pressure, volume, and work. Although the MCAT will test entropy from a thermodynamic rather than probabilistic understanding, we will briefly discuss the statistical model of entropy because it clarifies much of the confusion that arises from a characterization of entropy as a measure of “disorder.”

This chapter reviews the laws of thermodynamics with a specific focus on the zeroth, first, and second laws. We will examine how the zeroth law leads to the formulation of temperature scales. Thermal expansion will be discussed as an example of the relationship between thermal energy and physical properties like length, volume, and conductivity. We will then take a moment to examine thermodynamic terminology and functions that are also intimately related to the concepts discussed in Chapter 7 of MCAT General Chemistry Review. In the context of the first law—the conservation of energy—we will discuss the relationship between internal energy, heat, and work, and characterize specific heat and heat of transformation. We will also review the various processes by which a system goes from one equilibrium state to another and connect work, discussed in Chapter 2 of MCAT Physics and Math Review, with heat. Finally, we will investigate the second law of thermodynamics through the concepts of entropy and its measurement. The third law of thermodynamics is not directly tested on the MCAT, so we will only briefly mention it.

3.1 Zeroth Law of Thermodynamics

The zeroth law of thermodynamics is based on a simple observation: when one object is in thermal equilibrium with another object, say a cup of warm tea and a metal stirring stick, and the second object is in thermal equilibrium with a third object, such as your hand, then the first and third object are also in thermal equilibrium. As such, when brought into thermal contact, no net heat will flow between these objects. Note that thermal contact does not necessarily imply physical contact, as objects can be in thermal contact across space.

KEY CONCEPT

The zeroth law of thermodynamics states the transitive property in thermal systems: If a = b and b = c, then a = c.

TEMPERATURE

The formulation of the zeroth law—that no net heat flows between objects in thermal equilibrium, and the corollary that heat flows between two objects not in thermal equilibrium—actually arose from studies of temperature. At any given time, all substances have a particular temperature. In everyday language, we use the term temperature to describe qualitatively how hot or cold something is, but in thermodynamics, it has a more precise meaning. At the molecular level, temperature is proportional to the average kinetic energy of the particles that make up the substance. At the macroscopic level, it is the difference in temperature between two objects that determines the direction of heat flow. When possible, heat moves spontaneously from materials that have higher temperatures to materials that have lower temperatures. Heat itself refers to the transfer of thermal energy from a hotter object with higher temperature (energy) to a colder object with lower temperature (energy). If no net heat flows between two objects in thermal contact, then we can say that their temperatures are equal and they are in thermal equilibrium.

KEY CONCEPT

Temperature is a physical property of matter related to the average kinetic energy of the particles. Differences in temperature determine the direction of heat transfer.

Since the 18th century, scales have been developed to quantify the temperature of matter with thermometers. Some of these systems are still in common use, including the Fahrenheit (°F), Celsius (°C), and Kelvin (K) scales. Fahrenheit and Celsius are the oldest scales still in common use and are relatively convenient because they are based on the phase changes for water, as shown in Table 3.1. In the Celsius scale, 0° and 100° define the freezing and boiling temperatures of water. In the Fahrenheit scale, these phase change temperatures are defined as 32° and 212°.

 

°F

°C

K

Absolute Zero

−460

−273

0

Freezing Point of Water

32

0

273

Boiling Point of Water

212

100

373

Table 3.1. Relevant Points in the Three Major Temperature Scales

The Kelvin scale is most commonly used for scientific measurements and is one of the seven SI base units. It defines as the zero reference point absolute zero, the theoretical temperature at which there is no thermal energy, and sets the freezing point of water as 273.15 K. The third law of thermodynamics states that the entropy of a perfectly-organized crystal at absolute zero is zero. Note that there are no negative temperatures on the Kelvin scale because it starts from absolute zero. Although the Kelvin and Celsius scales have different zero reference points, the size of their units is the same. That is to say, a change of one degree Celsius equals a change of one unit kelvin. Because there are 180 degrees between water’s phase changes on the Fahrenheit scale, rather than 100 degrees as on both the Celsius and the Kelvin scales, the size of the Fahrenheit unit is smaller. The following formulas can be used to convert from one scale to another:

Equation 3.1

where FC, and K are the temperatures in Fahrenheit, Celsius, and Kelvin, respectively.

MCAT EXPERTISE

The only time Fahrenheit is used routinely on the MCAT is for body temperature, which is 98.6°F or 37°C. In the rare occasion that it is used for a quantitative analysis question, conversions will be given.

Example:

If a meteorologist says that the temperature will reach a high of 86°F today, what will be the high temperature in °C and in K?

Solution:

To convert from Fahrenheit to Celsius, use:

Now convert from Celsius to Kelvin:

K = C + 273 = 30 + 273 = 303 K

THERMAL EXPANSION

It has long been noted that some physical properties of matter change when the matter gets hotter or colder. Length, volume, solubility, and even the conductivity of matter change as a function of temperature. The relationship between temperature and a physical property of some matter was used to develop the temperature scales with which we are familiar today. For example, Daniel Fahrenheit developed the temperature scale that bears his name by placing a thermometer filled with mercury into a bath of ice, water, and ammonium chloride. The cold temperature caused the mercury to contract, and when the level in the glass tube stabilized at a lower level, he marked this as the zero reference on the scale. He then placed the same mercury thermometer in a mixture of ice and water (that is, at the freezing point for water). The slightly warmer temperature of this mixture caused the mercury to rise in the glass column, and when it stabilized at this higher level, Fahrenheit assigned it a value of 32°. When he stuck the thermometer under his (or someone else’s) tongue, he marked the even higher mercury level as 100° (not 98.6°). The details of how and why Fahrenheit came to choose these numbers (and the history of their adjustment since Fahrenheit first developed the scale) are beyond the scope of this discussion; rather, what is important to note is that a change in some physical property of one kind of matter—in this case, the height of a column of mercury—can be correlated to certain temperature markers, such as the phase changes for water. Once the scale has been set in reference to the decided-upon temperature markers, then the thermometer can be used to take the temperature of any other matter, in accordance with the zeroth law.

Because the property of thermal expansion was integral to the development of thermometers, let’s look a little more closely at this phenomenon. A change in the temperature of most solids results in a change in their length. Rising temperatures cause an increase in length, and falling temperatures cause a decrease in length. The amount of length change is proportional to the original length of the solid and the increase in temperature according to the equation

ΔL = αLΔT

Equation 3.2

where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature. The coefficient of linear expansion is a constant that characterizes how a specific material’s length changes as the temperature changes. This usually has units of K–1, although it may sometimes be quoted as °C–1. This difference is inconsequential because the unit size for the Kelvin and Celsius scales is the same.

REAL WORLD

It is because of thermal expansion that bridges and sidewalks have gaps between their segments; they allow for thermal expansion without damaging integrity.

MNEMONIC

When the temperature of an object changes, its length changes a lot (αLΔT).

Example:

A metal rod of length 2 m has a coefficient of linear expansion of 10−6 K−1. It is cooled from 1080°C to 80°C. What is the final length of the rod?

Solution:

By using the information given in the problem, we can substitute directly into the thermal expansion formula:

ΔL = αLΔT = (10−6 K)(2 m)(80 K − 1080 K) = −2 × 10−3 m

The negative sign represents a decrease in length. The original length was 2 m; therefore, the final length is 2 − (2 × 10−3) = 1.998 m.

Liquids also experience thermal expansion, but the only meaningful parameter of expansion is volume expansion. The formula for volumetric thermal expansion is applicable to both liquids and solids:

ΔV = βVΔT

Equation 3.3

where ΔV is the change in volume, β is the coefficient of volumetric expansion, V is the original volume, and ΔT is the change in temperature. The coefficient of volumetric expansion is a constant that characterizes how a specific material’s volume changes as the temperature changes. Its value is equal to three times the coefficient of linear expansion for the same material (β = 3α).

Example:

Suppose that a thermometer with 1 mL of mercury is taken from a freezer with a temperature of −25°C and placed near an oven at 225°C. If the coefficient of volume expansion of mercury is 1.8 × 10−4 K−1, by how much will the liquid expand?

Solution:

Use the information given to plug into the volumetric expansion formula:

MCAT Concept Check 3.1:

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

1.    What does the zeroth law of thermodynamics say?

2.    What is the maximum distance that two objects can be from one another and still adhere to the zeroth law of thermodynamics?

3.    How do the initial length of an object and the amount it expands for a given temperature change relate to one another?

4.    True or False: The Kelvin scale is the most accurate measurement method for temperature because it is based on absolute zero.