Question

Suppose you have a fixed amount of an ideal gas at a constant volume. If the pressure of the gas is doubled while the volume is held constant, what happens to its temperature? [Section 10.4]

Short answer

The temperature of an ideal gas remains the same when its pressure is doubled while its volume is held constant, as T2 = T1.

Step-by-step solution

Write down the initial and final conditions of the gas

We are given that the pressure of an ideal gas doubles while its volume is held constant. Let's denote the initial pressure as P1 and the final pressure as P2. Since the pressure doubles, P2 = 2 * P1. Let's denote the initial temperature as T1 and the final temperature as T2, which is what we need to find.

Apply the ideal gas law to the initial and final conditions

According to the ideal gas law, PV = nRT. For the initial condition, we have P1 * V = n * R * T1. For the final condition, we have P2 * V = n * R * T2. Since the volume, number of moles, and gas constant are the same for both conditions, we can write V = V, n = n, and R = R.

Solve for the final temperature in terms of the initial temperature

Let's now express the ideal gas law for the final condition in terms of the initial condition. We have (2 * P1) * V = n * R * T2. Now we can substitute P1 * V = n * R * T1 into this equation: (2 * P1) * V = 2 * (n * R * T1) => 2 * (P1 * V) = 2 * (n * R * T1). Dividing both sides by 2, we get: P1 * V = n * R * T1. Now, we can solve for the final temperature T2: T2 = T1.

Conclude the change in temperature

The final temperature T2 is equal to the initial temperature T1 after the pressure of an ideal gas is doubled while its volume is held constant. This means that when the pressure of an ideal gas doubles at constant volume, the temperature of the gas remains the same.