Physics 4311: Thermal Physics – HW solution 6

due date: Wednesday, Mar 11, 2026

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Problem 1: Diesel engine (5 points)

a)

adiabatic compression, $T_{i} = 300$ K

T_{f} = T_{i} {(\frac{V_{i}}{V_{f}})}^{γ - 1} = 994 K

b)

$T_{f}$ is above the ignition temperature of Diesel-air mixtue

a)

𝑝𝐴 = 𝑀𝑔 \Rightarrow p = \frac{𝑀𝑔}{A}

b)

\begin{array}{rcl} p V_{0} & = & 𝑝𝐴 z_{o} = N k_{B} T_{0} \\ 𝑝𝑉 & = & 𝑝𝐴𝑧 = N k_{B} T \end{array}

T ∕ T_{0} = z ∕ z_{0} \Rightarrow T = 2 T_{0}

c)

pressure stays constant

- Δ W = \int 𝑝𝑑𝑉 = (𝑀𝑔 ∕ A) Δ V = (𝑀𝑔 ∕ A) A z_{0} = 𝑀𝑔 z_{0}

d)

Δ U = \frac{3}{2} N k_{B} T - \frac{3}{2} N k_{B} T_{0} = \frac{3}{2} N k_{B} T_{0}

e)

Δ Q = Δ U - Δ W = \frac{3}{2} N k_{B} T_{0} + 𝑀𝑔 z_{0} = \frac{3}{2} N k_{B} T_{0} + N k_{B} T_{0} = \frac{5}{2} N k_{B} T_{0}

a)

𝑚𝑔 + p_{0} A = 𝑝𝐴 \Rightarrow p = p_{0} + \frac{𝑚𝑔}{A}

b)

infinitesimal displacement $𝑑𝑧$ , volume change $𝑑𝑉 = 𝐴𝑑𝑧$ , use adiabatic equation $p V^{γ - 1} =$ const

\begin{array}{rcl} 𝑑𝑝 & = & - γ \frac{const}{V^{γ + 1}} 𝑑𝑉 = - 𝛾𝑝 \frac{𝑑𝑉}{V} = - γ \frac{p}{V} 𝐴𝑑𝑧 \\ 𝑑𝐹 & = & 𝐴𝑑𝑝 = - γ \frac{p}{V} A^{2} 𝑑𝑧 \end{array}

c)

$𝑑𝐹 \sim - 𝑑𝑧$ , spring constant $k = 𝛾𝑝 A^{2} ∕ V$ , frequency $ω^{2} = k ∕ m = 𝛾𝑝 A^{2} ∕ (𝑚𝑉)$
period $τ = 2 π ∕ ω$