Prove that the slope on a TS diagram of: (a) An isochoric curve is \(T / C_{V}\). (b) An isobaric curve is \(T / C_{P}\).

An isochoric process is a thermodynamic process where the volume remains constant. In simpler terms, the gas doesn't expand or contract. During an isochoric process, any heat added to the system will change its internal energy and temperature, but not its volume. The specific heat capacity at constant volume, denoted as \(C_V\), is the amount of heat required to raise the temperature of a unit mass of gas by one degree Celsius, while maintaining constant volume.
In an isochoric process, the change in internal energy \(dU\) is given by the equation:
\[ dU = C_V dT\]
Here, \(dT\) represents the change in temperature. Using this, we can find the change in entropy \(dS\) as:
\[ dS = \frac{C_V dT}{T} \]
So, the derivative of temperature with respect to entropy (which is the slope on a TS diagram) for an isochoric process is:
\[ \frac{dT}{dS} = \frac{T}{C_V} \]
This slope shows how the temperature changes in response to changes in entropy at constant volume.

An isobaric process is one where the pressure of the gas remains constant. Unlike the isochoric process, the volume can change in an isobaric process, meaning the gas can expand or contract. The specific heat capacity at constant pressure, denoted as \(C_P\), is the amount of heat needed to raise the temperature of a unit mass of gas by one degree Celsius at a constant pressure.
In an isobaric process, the heat added \(dQ\) is given by the equation:
\[ dQ = C_P dT \]
To find the change in entropy, the equation becomes:
\[ dS = \frac{C_P dT}{T} \]
By rearranging this equation, we get the slope of the isobaric curve on a TS diagram:
\[ \frac{dT}{dS} = \frac{T}{C_P} \]
This represents how temperature changes with entropy at constant pressure. The slope is determined by the specific heat capacity at constant pressure.

Specific heat capacity is a crucial concept in thermodynamics. It tells us how much heat is needed to raise the temperature of a unit mass of a substance by one degree Celsius. There are two types of specific heat capacities commonly discussed in thermodynamics:
- **Specific Heat Capacity at Constant Volume (\(C_V\))**: This is used for processes at constant volume. It measures the amount of heat needed to change the temperature without changing the volume. For ideal gases, \(C_V\) is a constant.
- **Specific Heat Capacity at Constant Pressure (\(C_P\))**: This is used for processes at constant pressure. It measures the amount of heat needed to change the temperature without changing the pressure.
The specific heat capacities at constant volume and constant pressure are related, but they are used in different contexts. In an isochoric process, \(C_V\) is utilized, whereas in an isobaric process, \(C_P\) is relevant. In summary, understanding these specific heat capacities helps us analyze how heat transfer influences changes in temperature, pressure, volume, and entropy in different thermodynamic processes.