Skip to content

Ch. 3 Problems

← Back to chapter　|　View solutions

Table of Contents

Basic

Medium

Basic

B-1. Calculating Carnot Efficiency

Calculate the maximum efficiency of a Carnot cycle operating between a hot reservoir at \(T_{\text{hot}} = 500\) K and a cold reservoir at \(T_{\text{cold}} = 300\) K. Also, calculate the efficiency when the cold reservoir is lowered to \(T_{\text{cold}} = 200\) K, and compare the two results.

→ View solution

Medium

M-1. Entropy of Coins

When \(N\) coins are tossed, the number of microstates \(\Omega\) corresponding to "exactly half showing heads" is given by the binomial coefficient \(\binom{N}{N/2}\). Calculate \(\Omega\) for the cases \(N = 4\) and \(N = 100\), and verify how overwhelmingly large the number of states with "exactly half heads" becomes as \(N\) increases.

Hint

For the case \(N = 100\), it is useful to apply Stirling's approximation \(\ln N! \approx N \ln N - N\).

→ View solution

M-2. Statistical Mechanical Definition of Temperature

When two systems (with energies \(E_1\), \(E_2\), and total energy \(E = E_1 + E_2\) held constant) are in thermal equilibrium, derive from the condition that the total entropy \(S_{\text{total}} = S_1(E_1) + S_2(E_2)\) is maximized that \(\frac{\partial S_1}{\partial E_1} = \frac{\partial S_2}{\partial E_2}\), i.e., \(T_1 = T_2\).

→ View solution