Draw a free-body diagram of the combined piston-block system while it rises at constant speed. Label all forces and their directions.

Derive an expression for the work $W_{\mathrm{on}}$ done on the gas while the piston rises through distance $d$. Express the result in terms of $m$, $g$, and $d$.

Derive an expression for the energy $Q$ transferred to the gas during the rise. Express the result in terms of $m$, $g$, and $d$.

Calculate $Q$ when $m=12.0,\mathrm{kg}$, $d=0.180,\mathrm{m}$, and $g=9.80,\mathrm{m,s^{-2}}$.

Determine the factor by which the energy transferred to the gas changes.

Justify the answer using the functional dependence obtained in part (c).

Draw a free-body diagram of the piston and remaining sand at an intermediate point in the expansion. Label the force exerted by the gas, the force due to atmospheric pressure, and the gravitational force on the piston-sand system.

Derive an expression for the ratio $\dfrac{T_2}{T_1}$ in terms of $P_1$, $P_2$, $V_1$, and $V_2$.

Determine whether $T_2$ is greater than, less than, or equal to $T_1$.

Justify the answer using the first law of thermodynamics and the relationship between the internal energy and temperature of a monatomic ideal gas.

Sketch a graph of pressure $P$ as a function of volume $V$ for the expansion from state $1$ to state $2$. Label both states. On the same axes, sketch the path for an isothermal expansion that begins at state $1$ and ends at volume $V_2$. Clearly indicate the relative positions of the two paths.

Compare the final pressure in this isothermal process with $P_2$.

Justify the comparison using the graph and an appropriate mathematical relationship.

Describe a procedure that the student could use to obtain sufficient data to determine how absolute pressure depends on gas volume. Include appropriate equipment, the measurements to be made, and steps that improve the reliability of the investigation.

Indicate the independent variable, the dependent variable, and two quantities that should be held constant.

Calculate the reciprocal volume $1/V$ for each trial.

Plot absolute pressure $P$ on the vertical axis as a function of reciprocal volume $1/V$ on the horizontal axis. Label both axes with units, use sensible linear scales, plot all six points, and draw a best-fit line.

Determine the slope of the best-fit line, including units.

Calculate the number of moles of gas trapped in the syringe.

Determine whether the entropy of the gas increases, decreases, or remains constant after the membrane is punctured.

Justify the answer without using equations. Base the reasoning on the distribution of the gas particles and the tendency of an isolated system to approach equilibrium.

Derive expressions for $T_f$ and $P_f$ in terms of the initial quantities. Use the first law of thermodynamics, the internal-energy relationship for a monatomic ideal gas, and the ideal gas law.

Verify that the mathematical results from part (c) are consistent with the entropy conclusion from parts (a) and (b). Refer to both the particle distribution shown and the physical meanings of temperature and pressure.

Determine whether the net work done on the gas during one complete cycle is positive or negative.

Calculate the net work done on the gas during one complete cycle.

Derive an expression for the change in internal energy from state $B$ to state $C$ in terms of the pressures and volumes at the two states. Begin with the ideal-gas law and the kinetic-energy model for a monatomic ideal gas.

Calculate the energy transferred to the gas by heating during process $B\rightarrow C$.

Sketch a graph of internal energy $U$ as a function of volume $V$ during process $B\rightarrow C$. Label the values of $U$ at states $B$ and $C$.

Justify the sign and magnitude of the net energy transferred to the gas by heating during the complete cycle.

Determine which sample occupies the larger volume.

Justify your answer using the graph and the ideal-gas model.

Draw particle diagrams for samples $X$ and $Y$ at $300\,\mathrm{K}$. Show the same number of particles in each diagram and represent the relative container volumes.

Derive an expression for the slope of a pressure-versus-temperature graph for a sealed rigid sample in terms of $n$, $R$, and $V$.

Calculate the ratio $V_Y/V_X$ using the slopes or the graph values at $300\,\mathrm{K}$.

Sketch a graph of $V$ as a function of $T$ for sample $X$ when it is allowed to expand at constant pressure from $300\,\mathrm{K}$ to $450\,\mathrm{K}$. Label the initial point and the final volume in terms of $V_X$.

Determine the factor by which the root-mean-square speed changes when sample $X$ is heated from $300\,\mathrm{K}$ to $450\,\mathrm{K}$.

Justify why the pressure can remain constant during this heating process even though the atoms move faster.

Describe a feasible procedure, including the equipment and the measurements needed to vary sample mass and measure temperature change.

Describe two important variables that should be controlled and how repeated trials should be used.

Describe a graphical analysis that would test the expected dependence of temperature change on mass.

Determine the heater power from the slope of the calibration graph.

Calculate the energy delivered in each trial using the calibration relationship.

Plot $Q$ as a function of $\Delta T$ for the six trials. Label both axes with units, use a sensible scale, and draw a best-fit line.

Calculate the specific heat of the sample from the slope of the graph in part (d).

Justify whether the plotted data are consistent with a temperature-independent specific heat over the measured range.

Determine the direction of net energy transfer by thermal processes immediately after the objects are placed in contact. Do not use equations.

Justify your answer using the graph and a microscopic description of thermal energy transfer. Do not use equations.

Determine how the total entropy of the isolated two-object system changes before equilibrium is reached.

Justify your answer using the second law and the distribution of energy. Do not use equations.

Derive an expression for $c_B$ in terms of $m_A$, $c_A$, $m_B$, the two initial temperatures, and the equilibrium temperature.

Calculate $c_B$ using the values obtained from the graph.

Determine the new equilibrium temperature if the mass of object $B$ is doubled while all initial temperatures and specific heats remain unchanged.

Compare the rates of energy transfer through the two materials for the same nonzero temperature difference.

Derive an expression for the slope of the graph in terms of thermal conductivity $k$, area $A$, and thickness $L$.

Calculate the thermal conductivity $k_X$.

Calculate the thermal conductivity $k_Y$.

Calculate the energy transferred through material $Y$ in $180\,\mathrm{s}$ when $\Delta T=40\,\mathrm{K}$.

Determine the factor by which the conduction rate through material $X$ changes if its area is halved and its thickness is doubled while $\Delta T$ is unchanged.

Justify the factor in part (e)(i) using functional dependence.

Verify that the graph intercept is physically consistent with the conduction model.

Determine which curve represents the sample at $600\,\mathrm{K}$.

Justify your answer using two features of the graph.

Draw particle diagrams for the two samples in equal-volume containers. Show equal numbers of particles and use velocity arrows to represent the relative speed distributions.

Derive the relationship between root-mean-square speed $v_{\mathrm{rms}}$ and absolute temperature $T$ for atoms of mass $m$.

Calculate the ratio of the root-mean-square speeds of the $600\,\mathrm{K}$ and $300\,\mathrm{K}$ samples.

Sketch a pressure-versus-temperature graph for either sample when the number of atoms and volume are constant. Mark the two temperatures and show the relative pressures.

Justify the pressure difference between the two states using atomic collisions with the container walls.