Thermodynamic Processes and Their Derivations Using First Principles
Constant Volume Process (Isochoric Process)
A constant volume process occurs when the volume remains unchanged during the process.
From the First Law of Thermodynamics:
dQ = dU + dW
Since work done in a constant volume process is given by:
dW = P dV = 0
Thus, the heat transfer in this process directly changes the internal energy:
dQ = dU
For an ideal gas,
dU = m C_v dT
where C_v is the specific heat at constant volume.
Constant Pressure Process (Isobaric Process)
In a constant pressure process, the pressure remains unchanged.
From the First Law of Thermodynamics:
dQ = dU + PdV
For an ideal gas,
dU = m C_v dT
and using the ideal gas equation:
dW = P dV = P (V_2 - V_1)
Applying the relation between specific heats:
dQ = m C_p dT
where C_p is the specific heat at constant pressure.
Polytropic Process
A polytropic process follows the equation:
PV^n = constant
Differentiating both sides:
P dV + V dP = 0
Applying the First Law:
dQ = dU + dW
Work done is given by:
W = (P_2 V_2 - P_1 V_1) / (1 - n)
For an ideal gas,
dU = m C_v dT
Heat transfer is then:
Q = (C_p - C_v) / (1 - n) * (P_2 V_2 - P_1 V_1)
Isothermal Process
An isothermal process occurs at constant temperature, meaning dT = 0 and thus internal
energy remains unchanged (dU = 0).
Using the First Law:
dQ = dW
Work done is obtained from:
W = ∫ P dV
Using the ideal gas equation, P = (nRT) / V, we get:
W = nRT ln(V_2 / V_1)
Since dQ = dW, heat transfer is:
Q = W = nRT ln(V_2 / V_1)
Isothermal Process (Hyperbolic)
For a hyperbolic process, the equation PV = C holds.
The derivation remains the same as the isothermal process, leading to:
W = P_1 V_1 ln(V_2 / V_1)
Isentropic Process (Adiabatic and Reversible Process)
For an isentropic process, entropy remains constant, and no heat transfer occurs (dQ = 0).
Using the First Law:
dU = -dW
For an ideal gas:
dU = m C_v dT
and work done is:
W = (P_2 V_2 - P_1 V_1) / (1 - γ)
where γ = C_p / C_v is the adiabatic index.
Using PV^γ = constant, we derive:
T_2 = T_1 * (V_1 / V_2)^(γ - 1)
which connects temperature and volume for an isentropic process.