Question

Consider a process in which an ideal gas changes from state 1 to state 2 in such a way that its temperature changes from $300\mathrm{K}$ to $200\mathrm{K}$. (a) Describe how this change might be carried out while keeping the volume of the gas constant. (b) Describe how it might be carried out while keeping the pressure of the gas constant. (c) Does the change in $\mathrm{\Delta }E$ depend on the particular pathway taken to carry out this change of state? Explain.

Short answer

(a) To carry out the change in temperature from 300 K to 200 K while keeping the volume constant, we need to adjust the pressure accordingly, such that, $P_1/T_1=P_2/T_2$. As the temperature decreases, we need to decrease the pressure in the same proportion to maintain constant volume. (b) To carry out the change in temperature from 300 K to 200 K while keeping the pressure constant, we need to adjust the volume accordingly, such that, $V_1/T_1=V_2/T_2$. As the temperature decreases, we need to decrease the volume in the same proportion to maintain constant pressure. (c) The change in internal energy (∆E) is dependent only on the change in temperature and not on the path taken, as it is a function of its temperature: $E=\frac{f}{2}nRT$. Therefore, ∆E depends on the difference T2 - T1 and does not depend on the pathway taken for the change of state.

Step-by-step solution

a) Constant Volume Process

To carry out the change in temperature from 300 K to 200 K while keeping the volume constant, we can control the pressure of the gas. Using the ideal gas law $PV=nRT$, and keeping V and n constant, we have $P\propto T$. Therefore, to achieve the desired change in temperature, we need to adjust the pressure accordingly, such that, $P_1/T_1=P_2/T_2$. As the temperature decreases, we need to decrease the pressure in the same proportion to maintain constant volume.

b) Constant Pressure Process

To carry out the change in temperature from 300 K to 200 K while keeping the pressure constant, we can control the volume of the gas. Using the ideal gas law $PV=nRT$, and keeping P and n constant, we have $V\propto T$. Therefore, to achieve the desired change in temperature, we need to adjust the volume accordingly, such that, $V_1/T_1=V_2/T_2$. As the temperature decreases, we need to decrease the volume in the same proportion to maintain constant pressure.

c) Dependence of ∆E on Pathway

For an ideal gas, any change in internal energy (∆E) is dependent only on the change in temperature and not on the path taken. To support this, consider that the internal energy of an ideal gas is a function of its temperature: $E=\frac{f}{2}nRT$, where f is the number of degrees of freedom. The change in internal energy can be written as $\mathrm{\Delta }E=\frac{f}{2}nR(T_2-T_1)$. The expression for ∆E does not depend on the pressure or volume, only on the initial and final temperatures (∆E depends on the difference T2 - T1). Therefore, the change in internal energy (∆E) does not depend on the pathway taken for the change of state, only on the change in temperature.

Key concepts

Constant Volume Process

During a constant volume process, the volume of the gas remains fixed while its temperature changes. In this type of process, adjustments are made to the pressure to maintain the constant volume as the gas undergoes temperature changes.

To understand how to keep volume constant, we can use the ideal gas law:

• $PV=nRT$, where:
  • $P$ is the pressure,
  • $V$ is the volume,
  • $n$ is the amount of gas in moles,
  • $R$ is the ideal gas constant,
  • $T$ is the temperature in Kelvin.

In this exercise, we know that volume $V$ and the number of moles $n$ stay constant. Therefore, by rearranging the ideal gas equation, we have $P\propto T$. This means, if we reduce the temperature from 300 K to 200 K, the pressure must also decrease, maintaining $P_1/T_1=P_2/T_2$.

• As temperature decreases, pressure must proportionally decrease.
• Constant volume processes highlight the direct relationship between gas pressure and temperature.

Constant Pressure Process

In a constant pressure process, the pressure stays the same while the temperature of the gas changes. Just like in a constant volume process, we utilize the ideal gas law to understand how to manage the variables.

Here, maintaining pressure constant means adjusting the gas volume as its temperature changes:

• Using the ideal gas law $PV=nRT$:
  • Pressure $P$ is kept constant,
  • Thus, $V\propto T$

To decrease the temperature from 300 K to 200 K, we must reduce the volume proportionally, ensuring that $V_1/T_1=V_2/T_2$.

In simple terms:

• As temperature decreases, volume decreases to keep pressure steady.
• This change conveys the direct relationship between gas volume and temperature when pressure is constant.

Change in Internal Energy

The change in internal energy of an ideal gas, represented as $\mathrm{\Delta }E$, is an important concept independent of the pathway taken. The main determinant for $\mathrm{\Delta }E$ is the temperature change between the initial and final states.

For ideal gases, the internal energy $E$ is solely dependent on temperature:$$E=\frac{f}{2}nRT$$

• Where $f$ is the degrees of freedom of the gas.
• $n$ is the number of moles.
• $T$ is the temperature in Kelvin.

Given this, the change in internal energy $\mathrm{\Delta }E$ as the gas moves from state 1 to state 2 can be written as:$$\mathrm{\Delta }E=\frac{f}{2}nR(T_2-T_1)$$

• This illustrates that $\mathrm{\Delta }E$ is solely a function of temperature difference $T_2-T_1$.
• It demonstrates independence from the specific paths taken during changes in pressure or volume.

Understanding that internal energy change relies on temperature change helps clarify gas behaviors across different processes.