Ash: Import converted Year 1 Semester 2 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc

2026-09-11T14:49:18Z

Import converted Year 1 Semester 2 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc

New page

<div class="pn-lecture-meta">
<b>Lecture 10: Thermal Physics IV: Thermodynamic Transitions</b><br>
Thu 19 Feb 2026 13:00
</div>

[[Year 1 Sem 2/LC Temperature and Matter/Lecture 09|← Previous lecture]] · [[Year 1 Sem 2/LC Temperature and Matter|Module]] · [[Year 1 Sem 2/LC Temperature and Matter/Lecture 11|Next lecture →]]

== Thermodynamic Transitions ==

=== Isothermal (Fixed Temperature) ===

This may look like:

<ul>
<li>

A curve on a PV plot. As <nowiki>$PV = nRT$</nowiki>, a PV plot shows a <nowiki>$1/x$</nowiki> relationship for constant <nowiki>$T$</nowiki>.

</li>
<li>

A vertical straight line on a PT plot.

</li>
<li>

A vertical straight line on a VT plot.

</li>
</ul>

If the temperature change is <nowiki>$0$</nowiki>, <nowiki>$\Delta U$</nowiki> (which we derived last time only depends on temperature) follows: <div class="pn-import-equation"><nowiki>$$\Delta T = \Delta U = 0$$</nowiki></div>

Hence: <div class="pn-import-equation"><nowiki>$$0 = Q_\text{in} + W_\text{on}$$</nowiki></div>

=== Isochoric (Fixed Volume) ===

This may look like:

<ul>
<li>

A vertical straight line on a PV plot.

</li>
<li>

A proportional relationship on a PT plot.

</li>
<li>

A horizontal straight line on a VT plot.

</li>
</ul>

For an isochoric transition, <nowiki>$\Delta U = Q_in$</nowiki>, as: <div class="pn-import-equation"><nowiki>$$W_\text{on} = (-) \int P dV$$</nowiki></div>

And as <nowiki>$dV = 0$</nowiki>, <nowiki>$W_\text{on} = 0$</nowiki>

=== Isobaric (Fixed Pressure) ===

This may look like:

<ul>
<li>

A horizontal straight line on a PV plot.

</li>
<li>

A horizontal straight line on a PT plot.

</li>
<li>

A proportional relationship on a VT plot.

</li>
</ul>

For an isobaric transition, <nowiki>$P$</nowiki> is just a constant, so the integral: <div class="pn-import-equation"><nowiki>$$W_\text{on} = \int_{V_1}^{V_2} P \, dV = \left[PV\right]_{v_1}^{v_2}$$</nowiki></div>

Where <nowiki>$P$</nowiki> is constant, which is nice and easy. Generally: <div class="pn-import-equation"><nowiki>$$W_\text{on} = - \int P dV = (-) P \Delta V$$</nowiki></div>

<div class="pn-import-equation"><nowiki>$$\Delta U = \Delta Q - P \Delta V$$</nowiki></div>

== Heat Capacities of Ideal Gases ==

Recall that: <div class="pn-import-equation"><nowiki>$$\Delta Q = C \Delta T \implies C = \frac{dQ}{dT}$$</nowiki></div>

At a fixed volume: <div class="pn-import-equation"><nowiki>$$C_V = \left(\frac{dQ_\text{in}}{dT}\right)_T$$</nowiki></div>

And at a fixed pressure: <div class="pn-import-equation"><nowiki>$$C_P = \left(\frac{dQ_\text{in}}{dT}\right)_P$$</nowiki></div>

Mayer’s Relation says that: <div class="pn-import-equation"><nowiki>$$C_P = C_V + nR$$</nowiki></div>

This is an examinable derivation.

=== Deriving Mayer’s Relation ===

<div>

<i>Proof.</i> Consider a gas piston. We have state variables <nowiki>$V, P, T$</nowiki>, and <nowiki>$N$</nowiki> is constant as the piston is sealed. The first law of thermodynamics says: <div class="pn-import-equation"><nowiki>$$\Delta U = Q_\text{in} + W_\text{on}$$</nowiki></div>

We can consider this for both a constant volume or a constant pressure. For a constant volume: <div class="pn-import-equation"><nowiki>$$\Delta U = \Delta Q_1$$</nowiki></div>

And for a constant pressure: <div class="pn-import-equation"><nowiki>$$\Delta U = \Delta Q_2 - P \Delta V$$</nowiki></div>

Noting that the two <nowiki>$Q$</nowiki>s are different as we have two different transitions. We can create a slightly different expression for constant volume by multiplying by <nowiki>$\Delta T / \Delta T$</nowiki>: <div class="pn-import-equation"><nowiki>$$\Delta U = \frac{\Delta Q_1}{\Delta T} \times \Delta_T = C_V \Delta T$$</nowiki></div>

And for constant pressure: <div class="pn-import-equation"><nowiki>$$\Delta Q_2 = \Delta U + P \Delta V = \frac{\Delta Q_2}{\Delta T} \Delta T = C_P \Delta T$$</nowiki></div>

Hence: <div class="pn-import-equation"><nowiki>$$C_P \Delta T = \Delta U + P \Delta V$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$C_V \Delta T = \Delta U$$</nowiki></div>

Setting equal to each other based on <nowiki>$\Delta U$</nowiki>: <div class="pn-import-equation"><nowiki>$$\Delta U = \boxed{C_V \Delta T = C_P \Delta T - P \Delta V}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$C_P \Delta T - C_V \Delta T = P \Delta V$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\Delta T \left(C_P - C_V\right) = P \Delta V$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\implies C_P - C_v = \frac{\Delta V}{\Delta T} \times P \qquad \text{NB: $P$ is still constant.}$$</nowiki></div>

Using <nowiki>$PV = nRT$</nowiki>, for constant pressure we have <nowiki>$\frac{P \Delta V}{\Delta T} = nR$</nowiki>. Hence: <div class="pn-import-equation"><nowiki>$$C_P - C_V = nR$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$C_P = C_V + nR$$</nowiki></div> ◻

</div>

It is clear that <nowiki>$C_P > C_V$</nowiki>.

Note: We tend to use heat capacity at a constant volume unless specified.

== Back to Thermodynamic Transitions ==

=== Adiabatic Transitions (No Heat Transfer) ===

These seem quite a bit like isothermal transitions, but look different on a PV plot:

[[File:TM-Y1S2-lec10-01.png|thumb|center|600px|Differences in transitions on a PV Plot]]

[[File:TM-Y1S2-lec10-02.png|thumb|center|600px|Adiabatic Transitions on a PV and VT Plot]]

For isothermal transitions, <nowiki>$PV = \text{const}$</nowiki>.

For adiabatic transitions, <nowiki>$PV^\gamma = \text{const}$</nowiki> where <nowiki>$\gamma = \frac{C_P}{C_V}$</nowiki>

In a PV plane, adiabatic processes appear steeper than isothermal processes and adiabatic processing are steeper in the PV plane than the VT plane.

For an adiabatic transition, we can reduce the first law of thermodynamics like so: <div class="pn-import-equation"><nowiki>$$\Delta U = Q_\text{in} + W_\text{on}$$</nowiki></div>

<nowiki>$Q_\text{in} = 0$</nowiki> as there is no heat transfer, hence: <div class="pn-import-equation"><nowiki>$$\Delta U = W_\text{on}$$</nowiki></div>

=== Derivations ===

<div>

<i>Proof.</i> For adiabatic transitions, where <nowiki>$Q = 0$</nowiki>.

The First Law becomes <nowiki>$\Delta U = Q_\text{in} + W_\text{on}$</nowiki>. Since <nowiki>$Q_\text{in} = 0$</nowiki>: <div class="pn-import-equation"><nowiki>$$dU = 0 + dW_\text{on}$$</nowiki></div>

And assuming work is being done <i>on</i> the gas, the sign becomes negative: <div class="pn-import-equation"><nowiki>$$dU = -P dV$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\frac{dU dT}{dT} = -P dV$$</nowiki></div>

We slightly abuse the definition of heat capacity here, and this will be elaborated on in a later lecture: <div class="pn-import-equation"><nowiki>$$C_V dT = -P dV$$</nowiki></div>

Using <nowiki>$PV = nRT$</nowiki>, we have: <nowiki>$d(PV) = nR dT$</nowiki>: <div class="pn-import-equation"><nowiki>$$nR dT = P dV + V dP$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$dT = \frac{P dV + V dP}{nR}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$C_V dT = - PdV$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$0 = C_V \frac{P dV + V dP}{nR} + PdV$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$0 = C_V \left(P dV + V dP\right) + nR pdV$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$0 = P dV \left(C_v + nR\right) + C_V V dP$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$0 = P dV (C_P) + V dP (C_V)$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$0 = \frac{C_P}{C_V} PdV + V dP$$</nowiki></div>

Let <nowiki>$\gamma = \frac{C_P}{C_V}$</nowiki>, and dividing through by pressure and volume: <div class="pn-import-equation"><nowiki>$$\gamma \frac{dV}{V} + \frac{dP}{Ps}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\int \gamma \frac{dV}{V} = - \int \frac{dP}{P}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\gamma \ln V + c_1 = - \ln(P) + c_2$$</nowiki></div>

Finally: <div class="pn-import-equation"><nowiki>$$\gamma \ln(v) + \ln(p) = \text{const}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\ln(P V^\gamma) = \text{const}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$PV^\gamma = e^\text{const} = \text{another constant}$$</nowiki></div>

Hence: <div class="pn-import-equation"><nowiki>$$PV^\gamma = \text{const}$$</nowiki></div>

As required! ◻

</div>

Note that in a TV plane, we instead have: <div class="pn-import-equation"><nowiki>$$TV^{\gamma -1} = \text{const}$$</nowiki></div>

----
[[Year 1 Sem 2/LC Temperature and Matter/Lecture 09|← Previous lecture]] · [[Year 1 Sem 2/LC Temperature and Matter|Module]] · [[Year 1 Sem 2/LC Temperature and Matter/Lecture 11|Next lecture →]]

Year 1 Sem 2/LC Temperature and Matter/Lecture 10 - Revision history

Ash: Import converted Year 1 Semester 2 lecture notes from 443937a7d37850a3c57dfa863f35e80d07f2ccbc