Five grams of nitrogen gas at an initial pressure of 3.0 atm and at 20°C undergo an isobaric expansion until the volume has tripled. What is the gas temperature after the expansion (in °C)? The gas pressure is then decreased at constant volume until the original temperature is reached.

Verified step by step guidance

1. Step 1: Start by identifying the given values for the first part of the problem. The initial pressure \( P_1 = 3.0 \; \text{atm} \), the initial temperature \( T_1 = 20^\circ \text{C} = 293.15 \; \text{K} \) (convert to Kelvin by adding 273.15), and the volume triples during the isobaric expansion, so \( V_2 = 3V_1 \).
2. Step 2: Use the ideal gas law relationship for an isobaric process, where \( \frac{T_2}{T_1} = \frac{V_2}{V_1} \). Since \( V_2 = 3V_1 \), substitute this into the equation to find \( T_2 \): \( T_2 = 3T_1 \).
3. Step 3: Substitute \( T_1 = 293.15 \; \text{K} \) into the equation \( T_2 = 3T_1 \) to calculate \( T_2 \) in Kelvin. After finding \( T_2 \) in Kelvin, convert it back to Celsius using \( T_2(\text{°C}) = T_2(\text{K}) - 273.15 \).
4. Step 4: For the second part of the problem, the gas undergoes a process at constant volume where the pressure decreases until the original temperature \( T_1 = 293.15 \; \text{K} \) is reached. Use the ideal gas law relationship for an isochoric process, \( \frac{P_2}{P_1} = \frac{T_2}{T_1} \).
5. Step 5: Substitute the known values \( P_1 = 3.0 \; \text{atm} \), \( T_2 \) (from the first part), and \( T_1 = 293.15 \; \text{K} \) into the equation \( P_2 = P_1 \cdot \frac{T_1}{T_2} \) to calculate the final pressure \( P_2 \).

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. This law is fundamental in understanding the behavior of gases under various conditions, allowing us to calculate changes in temperature and pressure during processes like expansion and compression.

Isobaric Process

An isobaric process is a thermodynamic process in which the pressure remains constant while the volume and temperature of the gas change. In this scenario, as the nitrogen gas expands isobarically, its temperature increases, which can be calculated using the Ideal Gas Law, given the initial conditions and the final volume.

Charles's Law

Charles's Law states that the volume of a gas is directly proportional to its temperature (in Kelvin) when pressure is held constant. This principle is crucial for determining the final temperature of the gas after it has expanded, as it allows us to relate the initial and final states of the gas during the isobaric expansion.