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Chapter 23: Problem 43

\((a)\) A monatomic ideal gas initially at \(19.0^{\circ} \mathrm{C}\) is suddenly compressed to one-tenth its original volume. What is its temperature after compression? (b) Make the same calculation for a diatomic gas.

Short Answer

Expert verified
The final temperature for a monatomic gas is approximately 493 K. For a diatomic gas, it's approximately 553 K.

Step by step solution

01

Convert Temperature to Kelvin

The initial temperature is given in Celsius. It needs to be converted to Kelvin, as this is the unit of temperature in the International System of Units (SI). The conversion can be done using the formula \(T(K) = T(C) + 273.15\). Thus, \(T_0 = 19.0C + 273.15 = 292.15 K\).
02

Calculate the Final Temperature for a Monatomic Gas

The adiabatic equation \(T_{0}V_{0}^{\gamma - 1} = T_{1}V_{1}^{\gamma - 1}\) should be used to find the final temperature for a monatomic gas. Given the initial volume \(V_0\) is ten times the final volume \(V_1\), we can write the equation as \(T_{0}10^{\gamma - 1} = T_{1}\). Solving this equation for \(T_1\) yields \(T_1 = T_{0}10^{\gamma - 1}\). For a monatomic gas, \(\gamma = 5/3\). Substituting \(\gamma\) and \(T_0 = 292.15K\) into the equation yields the final temperature \(T_{1} = 292.15 K \times 10^{5/3 - 1} \approx 493 K\).
03

Calculate the Final Temperature for a Diatomic Gas

Use the same process for a diatomic gas. Only the value of \(\gamma\) changes to \(7/5\) in case of a diatomic gas. Substituting \(\gamma = 7/5\) and \(T_0 = 292.15K\) into the equation yields \(T_{1} = 292.15 K \times 10^{7/5 - 1}\), or approximately \(T_{1} \approx 553 K\).

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Most popular questions from this chapter

Calculate the rate at which heat would be lost on a very cold winter day through a \(6.2 \mathrm{~m} \times 3.8 \mathrm{~m}\) brick wall \(32 \mathrm{~cm}\) thick. The inside temperature is \(26^{\circ} \mathrm{C}\) and the outside temperature is \(-18^{\circ} \mathrm{C} ;\) assume that the thermal conductivity of the brick is \(0.74 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\)

A small electric immersion heater is used to boil \(136 \mathrm{~g}\) of water for a cup of instant coffee. The heater is labeled 220 watts. Calculate the time required to bring this water from \(23.5^{\circ} \mathrm{C}\) to the boiling point, ignoring any heat losses.

(a) One liter of gas with \(\gamma=1.32\) is at \(273 \mathrm{~K}\) and \(1.00 \mathrm{~atm}\) pressure. It is suddenly (adiabatically) compressed to half its original volume. Find its final pressure and temperature. \((b)\) The gas is now cooled back to \(273 \mathrm{~K}\) at constant pressure. Find the final volume. \((c)\) Find the total work done on the gas.

What mass of steam at \(100^{\circ} \mathrm{C}\) must be mixed with \(150 \mathrm{~g}\) of ice at \(0^{\circ} \mathrm{C}\), in a thermally insulated container, to produce liquid water at \(50^{\circ} \mathrm{C}\) ?

A quantity of ideal monatomic gas consists of \(n\) moles initially at temperature \(T_{1}\). The pressure and volume are then slowly doubled in such a manner as to trace out a straight line on the \(p V\) diagram. In terms of \(n, R\), and \(T_{1}\), find \((\) a) \(W,(b)\) \(\Delta E_{\text {int }}\), and \((c) Q .(d)\) If one were to define an equivalent specific heat for this process, what would be its value?
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