A constant volume process is also known as an isochoric process. An example is when heat is added to a gas in a container with fixed walls.

Because the walls can't move, the gas can not do work:

W = 0

In that case the First Law states:

Q = ΔE_int

The P-V diagram for this process is simple - it's a vertical line going up if heat is added, and going down if heat is removed.

In the case of a monatomic ideal gas:

E_int = (3/2)NkT = (3/2)nRT

Therefore Q = ΔE_int = (3/2) nRΔT

A constant temperature process is an isothermal process. An example is when a gas in a container that is immersed in a constant-temperature bath is allowed to expand slowly, or is compressed slowly.

If the temperature is constant there is no change in internal energy.

ΔE_int = 0

In that case the First Law states:

Q = W

The P-V diagram for this process follows an isotherm, a line of constant temperature.

In the case of an ideal gas at constant temperature, the pressure is inversely proportional to the volume:

P = nRT/V, so:

W = ∫ P dV = nRT ∫ (1/V) dV

The integral of 1/V is ln(V), and ln(A)-ln(B) = ln(A/B).

Therefore:

Q = W = nRT ln(V_f / V_i )

A constant pressure process is called an isobaric process. An example is a gas in a container sealed with a piston that is free to slide up and down.

If heat is added the temperature goes up and the system expands, so work is done.

The full First Law applies:

ΔE_int = Q - W

The P-V diagram for this process is a horizontal line, so the work done is simply:

W = P ΔV = nR ΔT

For a monatomic ideal gas:

ΔE_int = (3/2) nR ΔT

Plugging this into the First Law gives:

Q = ΔE_int + W

Q = (3/2) nR ΔT + nR ΔT = (5/2)nR ΔT