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Chapter 21: Problem 18

A vertical cylinder with a heavy piston contains air at a temperature of \(300 \mathrm{K}\). The initial pressure is \(200 \mathrm{kPa},\) and the initial volume is \(0.350 \mathrm{m}^{3} .\) Take the molar mass of air as \(28.9 \mathrm{g} / \mathrm{mol}\) and assume that \(C_{V}=5 R / 2 .\) (a) Find the specific heat of air at constant volume in units of \(\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\) A vertical cylinder with a heavy piston contains air at a temperature of \(300 \mathrm{K}\). The initial pressure is \(200 \mathrm{kPa},\) and the initial volume is \(0.350 \mathrm{m}^{3} .\) Take the molar mass of air as \(28.9 \mathrm{g} / \mathrm{mol}\) and assume that \(C_{V}=5 R / 2 .\) (a) Find the specific heat of air at constant volume in units of \(\mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\)

Short Answer

Expert verified
After performing the calculations, the specific heat of air at constant volume (c_v) in units of J/kg*C will be obtained.

Step by step solution

01

Analyzing the Given Data

Firstly, recall two important constants: The molar gas constant \(R = 8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1}\) and the molar mass of air \( M = 28.9 \times 10^{-3} \, \text{kg mol}^{-1} \). From the exercise, it's stated that \( C_v = \frac{5}{2}R \).
02

Perform Unit Conversion

We know that \( C_v = \frac{5R}{2} \), but we need to change it in terms of J/kg*C. So to convert the molar specific heat at constant volume (\( C_v \)) to the specific heat at constant volume (in J/kg*C), we divide by the molar mass of the substance (in this case air). So, the specific heat \( c_v \) (in J/kg*C) is given by \( c_v = \frac{C_v}{M} \)
03

Substituting the values

Let's substitute the values into the equation from step 2 to calculate the specific heat at constant volume. So, \( c_v = \frac{ \frac{5R}{2}}{M} \)
04

Calculate the final value

Upon substituting the known values into the formular \( c_v = \frac{ 5(8.314 \, \text{J} \, \text{mol}^{-1} \, \text{K}^{-1})/2}{28.9 \times 10^{-3} \, \text{kg mol}^{-1}} \), we can compute the specific heat \( c_v \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermodynamics
Thermodynamics is the study of heat, energy, and work. It helps us understand how energy is transferred and transformed in systems. In thermodynamics, energy is measured in joules and can be present in different forms, such as kinetic and potential energy. When dealing with gases, we often refer to the concepts of pressure, volume, and temperature to describe the state of the gas. These variables are interconnected, and changes in one can affect the others. For example, if you increase the pressure of a gas in a fixed volume, its temperature might also increase. Thermodynamic processes can be classified through various principles like constant volume (isochoric), constant pressure (isobaric), and others. In practical applications, understanding these processes allows us to develop systems like engines and refrigerators.
Molar Mass of Air
Molar mass is an essential concept in chemistry and physics. It is the mass of one mole of a substance, typically measured in grams per mole (g/mol). One mole of a substance contains Avogadro's number of particles, which is approximately 6.022 x 10^23 particles. For air, which is a mixture primarily composed of nitrogen and oxygen, the average molar mass is about 28.9 g/mol. This average accounts for the varying proportions of different gases in the atmosphere. Knowing the molar mass of air is crucial when converting between moles and grams. For instance, it helps us in calculations where we need to determine the movement of gases in and out of systems, how gases react under certain conditions, or when converting specific heat from a per mole to a per kilogram basis.
Ideal Gas Constant
The ideal gas constant is a fundamental constant in the equation of state for ideal gases. It is denoted by R and has the value of 8.314 J/mol*K. This constant allows us to relate pressure, volume, and temperature in the famous ideal gas equation: PV = nRT, where P is pressure, V is volume, n is the number of moles, and T is temperature in Kelvin. The ideal gas constant is vital because it anchors our calculations around how lots of gases behave under different conditions, assuming they behave ideally. While real gases deviate slightly from this behavior at high pressures or low temperatures, the ideal gas law provides a good approximation for understanding gas dynamics in many situations. It assists us in predicting how gases will adjust when subjected to changes in their environment, which is crucial for various industrial and scientific applications.

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Most popular questions from this chapter

(a) Show that the speed of sound in an ideal gas is $$v=\sqrt{\frac{\gamma R T}{M}}$$ where \(M\) is the molar mass. Use the general expression for the speed of sound in a fluid from Section \(17.1,\) the definition of the bulk modulus from Section \(12.4,\) and the result of Problem 59 in this chapter. As a sound wave passes through a gas, the compressions are either so rapid or so far apart that thermal conduction is prevented by a negligible time interval or by effective thickness of insulation. The compressions and rarefactions are adiabatic. (b) Compute the theoretical speed of sound in air at \(20^{\circ} \mathrm{C}\) and compare it with the value in Table \(17.1 .\) Take \(M=\) \(28.9 \mathrm{g} / \mathrm{mol} .\) (c) Show that the speed of sound in an ideal gas is $$v=\sqrt{\frac{\gamma k_{\mathrm{B}} T}{m}}$$ where \(m\) is the mass of one molecule. Compare it with the most probable, average, and rms molecular speeds.

A sealed cubical container \(20.0 \mathrm{cm}\) on a side contains three times Avogadro's number of molecules at a temperature of \(20.0^{\circ} \mathrm{C}\). Find the force exerted by the gas on one of the walls of the container.

From the Maxwell-Boltzmann speed distribution, show that the most probable speed of a gas molecule is given by Equation \(21.29 .\) Note that the most probable speed corresponds to the point at which the slope of the speed distribution curve \(d N_{v} / d v\) is zero.

In a constant-volume process, \(209 \mathrm{J}\) of energy is transferred by heat to 1.00 mol of an ideal monatomic gas initially at 300 K. Find (a) the increase in internal energy of the gas, (b) the work done on it, and (c) its final temperature.

A gas is at \(0^{\circ} \mathrm{C}\). If we wish to double the rms speed of its molecules, to what temperature must the gas be brought?
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