D. C. Pandey Solutions for Chapter: Thermometry, Thermal Expansion and Kinetic Theory of Gases, Exercise 6: Exercises

In case of hydrogen and oxygen at $NTP$, which of the following is the same for both?

A perfectly conducting vessel of volume, $V=0.4 {\mathrm{m}}^3$ contains an ideal gas at constant temperature, $T=273 \mathrm{K}$. A portion of the gas is let out and the pressure of the gas falls by, $\Delta p=0.24 \mathrm{atm}$. (Density of the gas at $\mathrm{STP}$ is $\rho =1.2 \mathrm{kg} {\mathrm{m}}^{-3}$ ). Find the mass of the gas which escapes from the vessel.

A thin-walled cylinder of mass, $m,$ height, $h$ and cross-sectional area, $A$ is filled with a gas and floats on the surface of water. As a result of leakage from the lower part of the cylinder, the depth of its submergence has increased by $\Delta h$. Find the initial pressure, $p_1$ of the gas in the cylinder if the atmospheric pressure is, $p_0$ and the temperature remains constant.

Find the minimum attainable pressure of an ideal gas in the process ,$T=T_0+\alpha V^2$, where, $T_0$ and $\alpha$ are positive constants and $V$ is the volume of one mole of gas.

A solid body floats in a liquid at a temperature, $t=50 °\mathrm{C}$ being completely submerged in it. What percentage of the volume of the body is submerged in the liquid after it is cooled to $t_0=0 °\mathrm{C}$, if the coefficient of cubic expansion for the solid is ${\gamma }_{\mathrm{s}}=0.3\times {10}^{-5} °{\mathrm{C}}^{-1}$ and of the liquid ${\gamma }_{\mathrm{l}}=8\times {10}^{-5 } °{\mathrm{C}}^{-1}$?

Two vessels connected by a pipe with a sliding plug contain mercury. In one vessel, the height of mercury column is $39.2 \mathrm{cm}$ and its temperature is $0 °\mathrm{C}$, while in the other, the height of mercury column is $40 \mathrm{cm}$ and its temperature is $100 °\mathrm{C}$. Find the coefficient of cubical expansion for mercury. The volume of the connecting pipe should be neglected.

Two steel rods and an aluminum rod of equal length, $l_0$ and equal cross-section are joined rigidly at their ends as shown in the figure below. All the rods are in a state of zero tension at $0 °\mathrm{C}$.

Find the length of the system when the temperature is raised to $\mathrm{\theta }$. Coefficient of linear expansion of aluminum and steel are ${\alpha }_{\mathrm{a}}$ and ${\alpha }_{\mathrm{s}}$, respectively. Young's modulus of aluminum is $Y_{\mathrm{a}}$ and of steel is $Y_{\mathrm{s}}$.

A metal rod, $A$ of $25 \mathrm{cm}$ length expands by, $0.050 \mathrm{cm}$ when, its temperature is raised from $0 °\mathrm{C}$ to $100 °\mathrm{C}$. Another rod, $B$ of a different metal of $40 \mathrm{cm}$ length expands by, $0.040 \mathrm{cm}$ for the same rise in temperature. A third rod, $C$ of $50 \mathrm{cm}$ length is made up of pieces of rods, $A$ and $B$ placed end to end, expands by $0.03 \mathrm{cm}$ on heating from $0 °\mathrm{C}$ to $50 °\mathrm{C}$. Find the lengths of each portion of the composite rod.