A barometer made of a very narrow tube (see Fig) is placed at normal temperature and pressure. The coefficient of volume expansion of mercury is 0.00018 per degree C and that of the tube is negligible. The temperature of mercury in the barometer is now raised by 1 degree C, but the temperature of the atmosphere does not change. Then the mercury height in the tube remains unchanged.

Step 1: Identify the given parameters.

The initial parameters and conditions are:

$$\gamma_{\text{Hg}} = 0.00018\,^{\circ}\text{C}^{-1}$$ $$\gamma_{\text{tube}} \approx 0$$ $$\Delta T = 1\,^{\circ}\text{C}$$ $$P_{\text{atm}} = \text{constant}$$

Step 2: State the initial equilibrium condition.

The atmospheric pressure balances the pressure of the mercury column of height $h$.

$$P_{\text{atm}} = \rho g h$$

Step 3: Determine the new density of mercury.

The density of mercury decreases as it expands with the temperature increase.

$$\rho' = \frac{\rho}{1 + \gamma_{\text{Hg}}\,\Delta T}$$

Step 4: Apply the equilibrium condition at the new temperature.

The atmospheric pressure remains constant, so the pressure of the new mercury column must equal the initial pressure.

$$\rho' g h' = \rho g h$$

Step 5: Express the new height of the mercury column.

Solve for the new height $h'$.

$$h' = h \frac{\rho}{\rho'}$$ $$h' = h(1 + \gamma_{\text{Hg}}\,\Delta T)$$

Step 6: Calculate the change in height.

Find the change in height $\Delta h$ using the standard height of $h = 76\text{ cm}$.

$$\Delta h = h' - h = h \gamma_{\text{Hg}}\,\Delta T$$ $$\Delta h = 76 \times 0.00018 \times 1$$ $$\Delta h = 0.0137\text{ cm}$$

Step 7: Evaluate the fractional change in height.

The fractional change in height depends only on the expansion coefficient and temperature change.

$$\frac{\Delta h}{h} = \gamma_{\text{Hg}}\,\Delta T = 0.018\%$$

Answer: The statement is false. The mercury height does not remain unchanged; it increases by $0.0137\text{ cm}$ (an increase of $0.018\%$).