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

When the pressure of an ideal gas is doubled at constant volume and fixed amount, its temperature also doubles. This can be derived from the modified Ideal Gas Law Equation \(P_1/T_1 = P_2/T_2\), which leads to the final temperature \(T_2 = 2T_1\).

Step-by-step solution

Ideal Gas Law Formula

The ideal gas law formula is given by: \(PV = nRT\), where: - P is the pressure in pascals (Pa), - V is the volume in cubic meters (m³), - n is the amount of gas in moles, - R is the gas constant (\(8.31 J/(mol·K)\)), - T is the temperature in kelvin (K). Since the volume V and the amount of the gas n are constant, we can write the equation as: \(P_1/T_1 = P_2/T_2\), where \(P_1\) and \(P_2\) are the initial and final pressures and \(T_1\) and \(T_2\) are the initial and final temperatures.

Given data

We are given: \(- P_1 = \) initial pressure \(- P_2 = 2P_1\) (pressure is doubled) \(- V = \) constant volume \(- n = \) fixed amount of gas We are supposed to find how the temperature changes when the pressure is doubled keeping the volume constant.

Using the modified Ideal Gas Law Equation

As we have already derived a modified equation for our specific conditions: \(P_1/T_1 = P_2/T_2\) Now we need to substitute the given data into the equation: \(P_1/T_1 = (2P_1)/T_2\)

Solving for T2

Now, we'll solve for \(T_2\) (the final temperature), which is what we are supposed to find out: \(\frac{P_1}{T_1} = \frac{2P_1}{T_2}\) \(\Rightarrow T_2 = 2T_1\) We find that the final temperature \(T_2\) is double the initial temperature \(T_1\). Thus, when the pressure of an ideal gas is doubled at constant volume and fixed amount, its temperature also doubles.

Key concepts

Pressure

Pressure in the context of gases relates to the force that the gas exerts on the walls of its container. This force is the result of gas molecules colliding with the container walls. Consider the gas molecules as tiny balls moving rapidly in all directions. Each collision with a wall contributes to what we measure as pressure.
When you increase the number of molecules or boost their speed (which happens if you increase temperature), you also increase the pressure.

• Pressure is measured in units called pascals (Pa).
• In our case, doubling the pressure means doubling the force per unit area exerted by the gas.
• Under the condition of constant volume, the ideal gas law helps us understand how changes in pressure affect other gas properties.

The ideal gas law formula, expressed as \(PV = nRT\), provides a clear relationship between pressure \(P\), volume \(V\), and temperature \(T\). This tells us how, by keeping some variables constant, we can study the effects on others.

Temperature

When we talk about temperature in relation to gases, it's a measure of the average kinetic energy of gas molecules. Think of temperature as a sign of how fast the gas molecules are moving. Higher temperature means faster-moving molecules. This not only impacts the energy but directly affects pressure in the context of constant volume.

• Temperature in the ideal gas law is measured in kelvin (K), which is a fundamental scientific unit that starts from absolute zero, the point where all molecular motion stops.
• As we saw in the exercise, when the pressure is doubled and volume stays the same, the temperature also doubles to maintain the balance dictated by the gas law.
• This happens because, at constant volume, increasing pressure requires an increase in the speed of molecules, resulting in a higher temperature.

Thus, if you ever need to predict how temperature changes under certain conditions, the ideal gas law is a reliable tool.

Constant Volume

Constant volume means that the gas is contained in a space that does not change in size. When the volume is constant, any change to pressure or temperature can be studied in isolation to better understand the relationships they have with each other.

• In constant volume, a change in pressure must be matched by a corresponding change in temperature, as no expansion or contraction in volume can occur to relieve any associated pressures.
• This concept plays a vital role in many practical applications, such as understanding how gas behaves within an engine cylinder where the volume is temporarily constant during certain phases of engine cycles.
• In our exercise, keeping volume constant simplified the problem to only involve temperature changes when pressure is altered.

Using the constant volume condition alongside the ideal gas law, we easily deduced that when pressure doubles, the temperature must too. Constant volume conditions sharpen our focus on how specific properties depend on others.