# 4.1.15. Thermodynamics¶

What is the First Law of Thermodynamics and what kind of processes is it applicable to?

What is the Second Law of Thermodynamics and what is it for a reversible process?

What is the entropy change for an ideal gas for a reversible and irreversible process?

How are the isentropic relations for an ideal gas derived from the entropy change?

## 4.1.15.1. What are examples of path functions and state variables?¶

Path functions: Heat and work (about the boundary)

State variables: Internal energy, enthalpy, entropy (about the fluid)

## 4.1.15.2. What is the First Law of Thermodynamics and what kind of processes is it applicable to?¶

(lower case implies per unit mass)

The first law applies to reversible and irreversible processes

## 4.1.15.3. What is the Second Law of Thermodynamics and what is it for a reversible process?¶

i.e. there is irreversible work done by internal molecular changes \(\Sigma\)

For a reversible process \(\Sigma = 0\)

i.e. \(ds = {dq \over T}\)

## 4.1.15.4. What is Enthalpy?¶

From the \(1^{st}\) law, for both reversible and irreversible processes:

where \(dw = -d(pv)\)

Enthalpy is a state variables

For both reversible and irreversible processes it equals the internal energy plus pressure-volume potential energy.

## 4.1.15.5. What is Entropy?¶

From the \(2^{nd}\) law for a reversible process:

Entropy is a macro state variable

For a reversible process it is the infinitisimal heat added over constant temperature (non-adiabatic).

## 4.1.15.6. What are the two property relations for entropy change from the 1st and 2nd laws of thermodynamics at constant pressure and constant volume?¶

1st Law (reversible and irreversible) \(\longrightarrow de = dw+dq\)

Reversible work done by system \(\longrightarrow dw = -pdv\)

Reversible heat added to system (2nd law) \(\longrightarrow dq = Tds\)

For reversible and irreversible processes (1) \(\longrightarrow Tds = de + pdv\)

Definition of enthalpy \(\longrightarrow dh = d(e+pv) = de + d(pv) = de + pdv + vdp\)

By rearrangement \(\longrightarrow de = dh - pdv - vdp\)

For reversible and irreversible processes (2) \(\longrightarrow Tds = dh - vdp\)

## 4.1.15.7. What is the entropy change for an ideal gas for a reversible and irreversible process?¶

\(ds = {de \over T}+{p \over T}dv\)

\(ds = {dh \over T} - {v \over T}dp\)

Calorically perfect

\(\longrightarrow de=c_vdT\)

\(\longrightarrow dh=c_pdT\)

Ideal gas

\(\longrightarrow p = {RT \over v}\)

\(\longrightarrow v = {RT \over p}\)

Hence

\(\longrightarrow s_2 - s_1 = c_v \int_1^2 {dT \over T} + R \int_1^2 {dv \over v} = c_v ln{T_2 \over T_1} + R ln{v_2 \over v_1} = c_v ln{T_2 \over T_1} + R ln{\rho_1 \over \rho_2}\)

\(\longrightarrow s_2 - s_1 = c_p \int_1^2 {dT \over T} - R \int_1^2 {dp \over p} = c_p ln{T_2 \over T_1} - R ln{p_2 \over p_1}\)

## 4.1.15.8. How are the isentropic relations for an ideal gas derived from the entropy change?¶

Isentropic \(\longrightarrow s_2 - s_1 = 0\)

Hence

\(\longrightarrow 0 = c_v ln{T_2 \over T_1} - R ln{\rho_2 \over \rho_1}\)

\(\longrightarrow 0 = c_p ln{T_2 \over T_1} - R ln{p_2 \over p_1}\)

Definition of \(c_v\) and \(c_p\):

\(c_v = {R \over {\gamma -1}}\)

\(c_v = {{\gamma R} \over {\gamma -1}}\)

Hence:

\(\longrightarrow ln{\rho_2 \over \rho_1} = {1 \over {\gamma -1}} ln{T_2 \over T_1}\)

\(\longrightarrow ln{p_2 \over p_1} = {{\gamma} \over {\gamma -1}} ln{T_2 \over T_1}\)

Hence:

\(\longrightarrow {\rho_2 \over \rho_1} = {T_2 \over T_1}^{1 \over {\gamma -1}}\)

\(\longrightarrow {p_2 \over p_1} = {T_2 \over T_1}^{{\gamma} \over {\gamma -1}}\)

Hence:

\(\longrightarrow {p_2 \over p_1} = {\rho_1 \over \rho_2}^{\gamma}\)

\(\longrightarrow {p \over {\rho^{\gamma}}} = \alpha\)