Question

One mole of an ideal gas is subjected to the following changes. Calculate the change in temperature for each case if \(C_{V, m}=3 R / 2\) a. \(q=-425 \mathrm{J}, w=185 \mathrm{J}\) b. \(q=315 .\) J, \(w=-315\) \(\mathbf{c} . q=0, w=225 \mathrm{J}\)

Short answer

The change in temperature for each case is: a. \(ΔT_{a} = -24.51 K\) b. \(ΔT_{b} = 25.24 K\) c. \(ΔT_{c} = -9.037 K\)

Step-by-step solution

List the given values

For all three cases, we have the following constants: Molar heat capacity at constant volume: \(C_{V,m} = \dfrac{3R}{2}\) Number of moles: 1 Ideal Gas Constant (R): 8.314 J/(mol·K)

Calculate change in internal energy

Using the first law of thermodynamics (\(ΔU = q - w\)), we can find the change in internal energy for each case. a. \(ΔU_{a} = (-425 J) - (185 J) = -610 J\) b. \(ΔU_{b} = (315 J) - (-315 J) = 630 J\) c. \(ΔU_{c} = (0 J) - (225 J) = -225 J\)

Calculate change in temperature using ideal gas equation

Now that we have the change in internal energy, we can find the change in temperature using the equation \(ΔU = nC_{V, m}ΔT\), where n is the number of moles, and \(C_{V,m}\) is the molar heat capacity at constant volume: a. \(ΔT_{a} = \dfrac{ΔU_{a}}{nC_{V, m}} = \dfrac{-610 J}{1 \cdot \dfrac{3}{2} \cdot 8.314} = -24.51 K\) b. \(ΔT_{b} = \dfrac{ΔU_{b}}{nC_{V, m}} = \dfrac{630 J}{1 \cdot \dfrac{3}{2} \cdot 8.314} = 25.24 K\) c. \(ΔT_{c} = \dfrac{ΔU_{c}}{nC_{V, m}} = \dfrac{-225 J}{1 \cdot \dfrac{3}{2} \cdot 8.314} = -9.037 K\) So, the change in temperature for each case is: a. \(ΔT_{a} = -24.51 K\) b. \(ΔT_{b} = 25.24 K\) c. \(ΔT_{c} = -9.037 K\)

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation connecting several physical properties of gases. It's expressed as \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature. This law helps us understand how gases behave under different conditions of pressure, volume, and temperature.
\( R \) is constant, making it easier for us to predict how changes in one property will affect the others. For example, if the volume remains constant and pressure increases, the temperature must increase as well to maintain the equality.

• Application: The Ideal Gas Law is often used when analyzing changes in conditions like in this exercise, where the properties of gases are modified due to heat and work interactions.
• Limitation: Keep in mind that the Ideal Gas Law assumes gases are perfect, but real gases show deviations at high pressures or low temperatures.

Molar Heat Capacity

Molar Heat Capacity describes how much heat one mole of a substance needs to change its temperature by one Kelvin. For gases, we typically talk about molar heat capacity at constant volume \( C_{V,m} \) or at constant pressure \( C_{P,m} \).
In this exercise, \( C_{V,m} = \frac{3}{2} R \) relates to measuring heat change without allowing the gas to expand (i.e., at constant volume).
This property is crucial in determining how much the temperature of the gas will change when there is a change in internal energy. In our calculations, we used \( C_{V, m} \) to relate internal energy changes directly to temperature changes.

• The units of molar heat capacity are usually J/(mol·K).
• Values of \( C_{P,m} \) and \( C_{V,m} \) can differ since gases tend to do work as they expand against external pressure.

Internal Energy

Internal Energy in a system refers to the total energy contained within the system, including kinetic and potential energies at the molecular level. Changes in internal energy are essential in thermal processes, especially in thermodynamics.
According to the First Law of Thermodynamics, the change in internal energy \( ΔU \) of a system can be expressed as the sum of heat transferred \( q \) to the system and the work done \( w \) by the system: \( ΔU = q - w \).

• Negative \( w \) indicates work done by the gas (expansion), causing energy to leave the system.
• Knowing \( ΔU \) helps evaluate temperature changes as represented in this exercise's solutions.

Internal energy changes tell us how heat absorbed or released results in temperature modification, often seen when evaluating ideal gas scenarios.

Change in Temperature

The Change in Temperature \( ΔT \) is a key outcome when analyzing heat and work interactions in gases. It helps quantify how much hotter or colder a gas becomes after undergoing energy exchanges.
The relationship between internal energy change and temperature change is derived using the formula \( ΔU = nC_{V,m}ΔT \). Here, the internal energy change \( ΔU \) effects \( ΔT \) directly depending on the moles present and \( C_{V,m} \).
In practical terms, this means:

• If a gas absorbs heat or work is done on it, temperature rises (positive \( ΔT \)).
• If it loses heat or does work on the surroundings, temperature drops (negative \( ΔT \)).

Temperature change gives an immediate sense of the thermal state adjustments in the system, pivotal in thermodynamic calculations like the one described.