An apparatus that liquefies helium is in a room maintained at 300 K. If the helium in the apparatus is at 4.0 K, what is the minimum ratio${\mathbf{Q}}_{\mathbf{to}}\mathbf{/}{\mathbf{Q}}_{\mathbf{from}}$, where${\mathbf{Q}}_{\mathbf{to}}$is the energy delivered as heat to the room and${\mathbf{Q}}_{\mathbf{from}}$is the energy removed as heat from the helium?

${\mathrm{Q}}_{\mathrm{to}}$is the energy delivered as heat to the room

${\mathrm{Q}}_{\mathrm{from}}$is the energy removed as heat from helium

Room temperature$T_{to}=300K$

The temperature of the system$T_{from}=4K$

By using the concept of change in entropy for a complete cycle is zero. i.e. entropy is a state function, by using the equation, we can find the ratio${\mathbf{Q}}_{\mathbf{t}\mathbf{o}}\mathbf{/}{\mathbf{Q}}_{\mathbf{f}\mathbf{r}\mathbf{o}\mathbf{m}}$.

Formula:

The entropy change using the second law of thermodynamics,

$∆\mathrm{S}=\frac{\mathrm{Q}}{\mathrm{T}}$ (1)

The net entropy change of the system due to energy transfer by the body and to the body can be given using equation (1) as:

$$\begin{gathered} ∆\mathrm{S}=∆{\mathrm{S}}_{\mathrm{to}}+∆{\mathrm{S}}_{\mathrm{from}} \\ =\frac{{\mathrm{Q}}_{\mathrm{to}}}{{\mathrm{T}}_{\mathrm{to}}}-\frac{{\mathrm{Q}}_{\mathrm{from}}}{{\mathrm{T}}_{\mathrm{from}}}...........................\left(2\right) \end{gathered}$$

$∆{\mathrm{S}}_{\mathrm{from}}$is negative because heat is removed from the substance.

As we know, entropy is a state function total change in entropy must be zero i.e.$∆\mathrm{S}=0$

Thus, the equation of entropy change gives the relation of energy and temperature as follows:

$\frac{{\mathrm{Q}}_{\mathrm{to}}}{{\mathrm{T}}_{\mathrm{to}}}=\frac{{\mathrm{Q}}_{\mathrm{from}}}{{\mathrm{T}}_{\mathrm{from}}}$

(where${\mathrm{Q}}_{\mathrm{to}}$ is the heat delivered to the room,${\mathrm{Q}}_{\mathrm{from}}$ is heat removed from the helium,${\mathrm{T}}_{\mathrm{to}}$ is the room temperature, and${\mathrm{T}}_{\mathrm{from}}$ is the helium temperature.)

$$\begin{gathered} \frac{{\mathrm{Q}}_{\mathrm{to}}}{{\mathrm{Q}}_{\mathrm{from}}}=\frac{300\mathrm{K}}{4\mathrm{K}} \\ =75.0 \end{gathered}$$

Hence, the required value of the ratio is 75.0