### Applying the ideal gas law

This exercise is a part of Educator Guide: Climate Change Spikes Baseball Homers / View Guide

Directions for teachers:

This discussion activity can be used to help students understand and apply the ideal gas law. Before beginning this work, students should know the ideal gas equation and its components. Have students discuss the first two sets of questions with a partner. Students should then read the Science News article “Baseball’s home run boom is due, in part, to climate change” and answer the last set. A version of the article, “Climate change spikes baseball homers,” appears in the May 6, 2023 & May 20, 2023 double issue of Science News.

#### Defining relationships from an equation

1. What is the ideal gas law equation, and what does it state? Name the components in the formula. Which of the components are variables?

The ideal gas law equation is PV=nRT.

The ideal gas law says that for an ideal gas, pressure (P) times volume (V) equals the number of moles of gas (n) times the ideal gas constant (R) times temperature (T). The variables are pressure, volume, number of moles, and temperature. R is a constant.

2. What is a direct relationship between two variables? How is an indirect, or inverse, relationship between variables different? Give examples.

When variables have a direct relationship, they change in the same direction. For example, if variables x and y have a direct relationship, when x increases, y also increases. In an indirect, or inverse, relationship, the variables go in opposite directions when they change. If variables x and y have an inverse relationship, when x increases, y will decrease.

3. Write the equation with all variables on one side and the ideal gas constant (R) on the other. Explain the relationships between the variables in the equation. How do the variables in the numerator relate to each other, if those in the denominator stay the same? What about the variables in the denominator, if those in the numerator stay the same? What happens when you change one variable from the numerator and allow one from the denominator to change? Hint: If these questions are difficult, put in numbers for each variable to test what happens.

PV/ nT = R is the equation. The two variables, P and V, in the numerator have an inverse relationship; if either P or V goes up, while n and T stay the same, the other will have to go down. The same is true of n and T in the denominator, if P and V stay the same. But if P and T stay the same and you change either V or n, the other one of the pair will have to change in the same direction, and we say the variables have a direct relationship. If V increases, then n also increases. If you change P or T instead, keeping n and V the same, you again see a direct relationship. If P goes up, T has to go up. If P goes down, T has to go down.

#### Explaining relationships using a simulation

1. Click the “Gas Properties” PhET simulation. Determine how changing P, V, n or T will influence other gas properties. Then change one variable and hold the other three constant to see what happens. What do you observe?

Students should see that the relationships defined by the equation are shown in the simulation.

2. What happens to the speed of the gas particles when you increase the temperature of the gas in the simulation? If you can, use an equation to explain this relationship.

As the temperature of a gas increases, the gas particles appear to move faster. That makes sense because if temperature increases, that means that the kinetic energy of the gaseous particles increases. Kinetic energy equals 1/2 mass times (velocity squared), and for a gas, kinetic energy equals 3/2 (ideal gas constant times temperature). Setting the two equations equal shows that as temperature increases, the kinetic energy of the gas increases, which corresponds to an increase in the velocity of the gas particles.

3. Gas pressure is defined as the force exerted by a gas per unit area on the surface of a container or another substance. Higher-velocity and more frequent collisions of gas particles with another substance increase the pressure of the system.

Pick two variables from the ideal gas equation and explain their relationship, stating how changing one will affect the environment and lead to a change in the other, when all other variables are constant. (Don’t just state the relationship like you did above.)

Student responses will vary based on the variables selected. If the amount of a gas and the volume of a container are held constant, an increase in temperature will increase the kinetic energy of the gas, which increases the velocity of the gas and the number of times the gas particles collide with the surface of the container, therefore increasing the pressure of the system.

#### Applications of the ideal gas law

1. The article “Baseball’s home run boom is due, in part, to climate change” focuses on the relationship between rising temperatures and increases in the number of home runs hit each year during the Major League Baseball season. The writer explained the roles temperature and air density played in increasing the number of home runs hit during the MLB season. Explain the relationship between temperature and air density using the ideal gas law.

As an extension, explain the relationship using the ideal gas law equation. Hint: Substitute mass of gas/molar mass of the gas (this fraction is the number of moles of the gas) for n and solve for mass of the gas/volume (this fraction is density). You should arrive at an expression that has temperature in the denominator.

As the temperature increases, the kinetic energy of the gas and its particle velocity will increase. If pressure stays the same, the air will expand in volume, which will decrease the air density. According to the equation, density of a gas equals pressure times molar mass divided by (the ideal gas constant times temperature). Therefore, temperature and density are inversely related.

2. What other factor decreases when air density drops? What principle explains that direct relationship?

When the air density drops, the air resistance also drops on the ball, so the ball will travel farther in the air before falling to the ground under the influence of gravity. This is because in less dense air, the baseball will come in contact with fewer air molecules, so fewer forces are slowing the baseball. Newton’s first law of motion about inertia states that “an object in motion will remain in motion, and an object at rest will remain at rest, unless acted upon by an outside force.” So with fewer forces slowing it, the ball can travel farther before it hits the ground, which means more home runs!

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