Samples consisting of \(1 \mathrm{~mol} \mathrm{~N}\) and \(1 \mathrm{~mol} \mathrm{} \mathrm{CH}_{4}\) are in identical but separate containers, with initial temperatures of 500. K. Both gases gain 1200 . J of heat at constant volume. Do the gases have the same final temperature? If not, which gas has the higher final temperature? Justify your reasoning.

When we talk about constant volume heat capacity (\( C_v \)), we're discussing how much heat energy is required to raise the temperature of a substance by 1 degree Celsius (or 1 Kelvin) when the volume remains unchanged. In a more technical sense, it's a measure of the energy storage of a gas per unit temperature change under the condition of constant volume.

For example, diatomic gases like nitrogen (\( N_2 \)) have a constant volume heat capacity of approximately \( \frac{5}{2}R \) (where \( R \) is the ideal gas constant). This is due to their molecular structure, which influences how they store energy. Similarly, more complex molecules like methane (\( CH_4 \)) have different heat capacities, typically \( \frac{3}{2}R \) for non-linear triatomic gases. This difference is crucial for understanding how different gases respond to heat addition in closed systems.

Key takeaway:

• \( C_v \) is an intrinsic property of a substance that can help us predict how the temperature of a gas will change when energy is added or removed.

How do we calculate the change in temperature of a gas when it's heated at constant volume? The formula to determine this is quite straightforward:\[ \Delta T = \frac{q}{nC_v} \]where \( \Delta T \) is the temperature change, \( q \) is the amount of heat transferred, \( n \) is the number of moles of gas, and \( C_v \) is the constant volume heat capacity.

In the exercise, we apply this equation to find out how much the temperature of the gas rises after adding 1200 J of heat to 1 mole of nitrogen and methane respectively. Since the gases have different heat capacities, the calculated temperature changes (\( \Delta T \)) will also be different, leading to different final temperatures even though the initial temperatures and heat added were the same.

Essential to remember:

• The heat capacity of the gas influences the outcome of the temperature change calculation, underscoring the importance of knowing this property to predict the behavior of gases.

The thermodynamics of gases is a vast and crucial area in physics that explains the behavior of gases under various conditions like temperature, volume, and pressure changes. It draws on concepts from the kinetic theory of gases and the laws of thermodynamics. In our context, we're mostly interested in how these concepts apply to heat transfer.

The exercise demonstrates a scenario in the thermodynamic study of gases where we consider the energy (heat) input to a system at a constant volume. It allows students to engage with such fundamental principles as the conservation of energy, relationship between heat and temperature, and the molecular properties of gases.

Integrating knowledge:

• Understanding the thermodynamics of gases, including constant volume processes, is fundamental in predicting how they will behave when subjected to thermal processes in closed systems.
• It ties together the microscopic molecular structure of gases with macroscopic observable quantities like temperature change, which can be particularly insightful in fields that span from engineering to meteorology.