Chapter 8, The second law and entropy Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

A building is maintained at a temperature $T$ by means of an ideal heat pump which uses a river at temperature $T_0$ as a source of heat. The heat pump consumes power $W$, and the building loses heat to its surroundings at a rate $\alpha\left(T-T_0\right)$. Show that $T$ is given by
$$
T=T_0+\frac{W}{2 \alpha}\left(1+\sqrt{1+4 \alpha T_0 / W}\right) .
$$

Shoukat Ali

Other Schools

01:46

Problem 2

A possible ideal-gas cycle operates as follows.
(i) from an initial state $\left(p_1, V_1\right)$ the gas is cooled at constant pressure to $\left(p_1, V_2\right)$.
(ii) the gas is heated at constant volume to $\left(p_2, V_2\right)$.
(iii) the gas expands adiabatically back to $\left(p_1, V_1\right)$.
Assuming constant heat capacities, show that the thermal efficiency is
$$
1-\gamma \frac{\left(V_1 / V_2\right)-1}{\left(p_2 / p_1\right)-1}
$$

Prem Bijarniya

Numerade Educator

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Problem 3

In an isothermal expansion of an ideal gas, an amount of work $W=\int p d V$ is done by the system on its surroundings. The internal energy $U$ of the gas does not change, because the system draws in heat $Q$ from a reservoir, and for an isothermal process in an ideal gas, $U$ remains constant. Since $\Delta U=0$ we have $Q=W$. So in this process, heat has been drawn from a single reservoir, and an equivalent amount of work has been done, 'turning' the heat totally into work. Is this a violation of the Kelvin statement of the second law?

Ankur S

Numerade Educator

02:40

Problem 4

Perpetual motion. A machine that could in principle operate for ever, in spite of the presence of friction or exhaust heat, is said to be capable of 'perpetual motion', and is called a perpetuum mobile from the Latin. This concept no longer features much in the study of thermal physics, but it played a useful role historically in getting clarity about what processes are and are not possible. Two types of perpetuum mobile may be distinguished. A perpetual motion machine of the first kind produces more energy that it uses. A perpetual motion of the second kind produces exactly as much energy as it uses, and keeps running indefinitely by recycling all the output energy, whether it is produced in the form of work or heat.
(i) Explain which of the laws of thermodynamics are broken by such machines. (ii) Show how a perpetuum mobile of the second kind could be constructed by combining two reversible heat engines of different efficiencies (if they could be found).

Shazia Naz

Numerade Educator