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Ch 19: Work, Heat, and the First Law of Thermodynamics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 19: Work, Heat, and the First Law of ThermodynamicsProblem 53b
Chapter 19, Problem 53b

A 10-cm-diameter cylinder contains argon gas at 10 atm pressure and a temperature of 50°C. A piston can slide in and out of the cylinder. The cylinder's initial length is 20 cm. 2500 J of heat are transferred to the gas, causing the gas to expand at constant pressure. What are the final length of the cylinder?

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1
Convert all given quantities to SI units. The diameter of the cylinder is 10 cm, so the radius is 5 cm or 0.05 m. The initial length of the cylinder is 20 cm or 0.2 m. The pressure is 10 atm, which can be converted to Pascals using the relation: \(1 \text{ atm} = 1.013 \times 10^5 \text{ Pa}\). The temperature is 50°C, which can be converted to Kelvin using \(T(K) = T(°C) + 273.15\).
Calculate the initial volume of the gas using the formula for the volume of a cylinder: \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the length of the cylinder. Substitute the values of \(r = 0.05 \text{ m}\) and \(h = 0.2 \text{ m}\) to find the initial volume.
Use the first law of thermodynamics, \(Q = W + \Delta U\), where \(Q\) is the heat added, \(W\) is the work done by the gas, and \(\Delta U\) is the change in internal energy. Since the process occurs at constant pressure, the work done by the gas is \(W = P \Delta V\), where \(P\) is the pressure and \(\Delta V\) is the change in volume. Rearrange the equation to solve for \(\Delta V\): \(\Delta V = \frac{Q}{P}\).
Add the change in volume \(\Delta V\) to the initial volume \(V_{\text{initial}}\) to find the final volume \(V_{\text{final}}\): \(V_{\text{final}} = V_{\text{initial}} + \Delta V\).
Determine the final length of the cylinder using the formula for the volume of a cylinder again: \(V = \pi r^2 h\). Rearrange to solve for the final length \(h_{\text{final}}\): \(h_{\text{final}} = \frac{V_{\text{final}}}{\pi r^2}\). Substitute the values of \(V_{\text{final}}\) and \(r\) to find the final length.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, temperature, and number of moles of a gas through the equation PV = nRT. In this scenario, understanding how the gas behaves under constant pressure is crucial for determining how its volume changes with temperature and heat transfer.
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Thermodynamics and Heat Transfer

Thermodynamics is the study of energy transfer, particularly heat. In this case, the 2500 J of heat added to the gas at constant pressure leads to an increase in volume, which can be calculated using the first law of thermodynamics, where the change in internal energy equals heat added minus work done by the system.
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Work Done by a Gas

When a gas expands against a piston, it does work on the piston, which can be calculated using the formula W = PΔV, where P is the pressure and ΔV is the change in volume. This concept is essential for determining how the gas's expansion affects the final length of the cylinder after heat is added.
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Related Practice
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n moles of an ideal gas at temperature T1 and volume V1 expand isothermally until the volume has doubled. In terms of n, T1, and V1, what is the final temperature?

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An ideal-gas process is described by p=cV1/2, where c is a constant. 0.033 mol of gas at an initial temperature of 150°C is compressed, using this process, from 300 cm3 to 200 cm3. How much work is done on the gas?

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Textbook Question

n moles of an ideal gas at temperature T1 and volume V1 expand isothermally until the volume has doubled. In terms of n, T1, and V1, what are the heat energy transferred to the gas?

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n moles of an ideal gas at temperature T1 and volume V1 expand isothermally until the volume has doubled. In terms of n, T1, and V1, what are the work done on the gas?

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