Define heat capacity. What are C_Pand C_V? Show that C_P – C_V = R.

Heat capacity of a substance is the amount of heat energy required to raise its temperature by one degree. There are two main types of heat capacities: heat capacity at constant volume (C_v) and heat capacity at constant pressure (C_p).

Definitions:

1. C_v: The molar heat capacity at constant volume, representing the heat required to increase the temperature of a system when its volume remains constant. C_v measures the change in internal energy (ΔU) with temperature under these conditions.
2. C_p: The molar heat capacity at constant pressure, indicating the heat required to raise the system's temperature while maintaining constant pressure. Here, the system expands, doing work on its surroundings, requiring additional heat. Thus, C_p is always greater than C_v.

Deriving C_p − C_v = R:

Using the First Law of Thermodynamics:

ΔU=q−W, where

ΔU is the change in internal energy,

q is the heat added, and

W is the work done by the system.

Process at Constant Pressure:

• When heat is absorbed at constant pressure, the system’s heat capacity is C_p, and q=nC_pΔT, where n is the number of moles and ΔT is the temperature change.
• Under constant pressure, the work done by an expanding gas is W=PΔV=nRΔT for an ideal gas, where R is the universal gas constant.

Process at Constant Volume:

At constant volume, there is no work done (W=0), so the heat added only increases internal energy: ΔU=nC_vΔT.

Relating C_p and C_v:

• Applying the First Law of Thermodynamics for a process at constant pressure, we get: ΔU=nC_pΔT−nRΔT
• Since ΔU=nCvΔT, we can set up the equation: nC_vΔT=nC_pΔT−nRΔT

Simplifying:

Dividing both sides by nΔT (assuming they are non-zero), we find: C_p − C_v = R

Conclusion: This relationship shows that for an ideal gas, the difference between the heat capacities at constant pressure and constant volume is equal to the universal gas constant:

C_p − C_v= R