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Chapter 9: Problem 9

Show that the volume expansion coefficient of an ideal gas is \(\beta=1 / T\), where \(T\) is the absolute temperature.

Short Answer

Expert verified
Question: Show that the volume expansion coefficient of an ideal gas is inversely proportional to the absolute temperature. Answer: The volume expansion coefficient (β) of an ideal gas is inversely proportional to the absolute temperature, which means β = 1 / T.

Step by step solution

01

Understand the volume expansion coefficient and ideal gas law

The volume expansion coefficient (β) of a substance is a measure of how the volume of the substance changes with temperature, at a constant pressure. Mathematically, it is defined as: \(\beta = \frac{1}{V} \frac{\partial V}{\partial T}\Big|_P\) where V is the volume, T is the absolute temperature, and P is the pressure. The ideal gas law relates the pressure (P), volume (V), and temperature (T) of an ideal gas: \(PV = nRT\) where n is the number of moles of the gas and R is the ideal gas constant.
02

Express the volume as a function of temperature and pressure

From the ideal gas law, we can express the volume (V) as a function of temperature (T) and pressure (P): \(V = \frac{nRT}{P}\)
03

Calculate the partial derivative of the volume with respect to the temperature

Now, we will calculate the partial derivative of the volume with respect to the temperature (keeping the pressure constant): \(\frac{\partial V}{\partial T}\Big|_P = \frac{\partial}{\partial T} \left(\frac{nRT}{P}\right)\Big|_P = \frac{nR}{P}\) Here, we treat n, R, and P as constants.
04

Calculate the volume expansion coefficient

Finally, we will plug our expression for the partial derivative of the volume with respect to the temperature back into the equation for the volume expansion coefficient: \(\beta = \frac{1}{V} \frac{\partial V}{\partial T}\Big|_P = \frac{1}{\frac{nRT}{P}} \frac{nR}{P}\) Simplifying, we get: \(\beta = \frac{1}{T}\) Therefore, the volume expansion coefficient of an ideal gas is inversely proportional to the absolute temperature, β = 1 / T.

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Most popular questions from this chapter

The side surfaces of a 3-m-high cubic industrial (?) furnace burning natural gas are not insulated, and the temperature at the outer surface of this section is measured to be \(110^{\circ} \mathrm{C}\). The temperature of the furnace room, including its surfaces, is \(30^{\circ} \mathrm{C}\), and the emissivity of the outer surface of the furnace is 0.7. It is proposed that this section of the furnace wall be insulated with glass wool insulation \((k=0.038 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) wrapped by a reflective sheet \((\varepsilon=0.2)\) in order to reduce the heat loss by 90 percent. Assuming the outer surface temperature of the metal section still remains at about \(110^{\circ} \mathrm{C}\), determine the thickness of the insulation that needs to be used. The furnace operates continuously throughout the year and has an efficiency of 78 percent. The price of the natural gas is \(\$ 1.10 /\) therm ( 1 therm \(=105,500 \mathrm{~kJ}\) of energy content). If the installation of the insulation will cost \(\$ 550\) for materials and labor, determine how long it will take for the insulation to pay for itself from the energy it saves.

A 4-m-long section of a 5-cm-diameter horizontal pipe in which a refrigerant flows passes through a room at \(20^{\circ} \mathrm{C}\). The pipe is not well insulated and the outer surface temperature of the pipe is observed to be \(-10^{\circ} \mathrm{C}\). The emissivity of the pipe surface is \(0.85\), and the surrounding surfaces are at \(15^{\circ} \mathrm{C}\). The fraction of heat transferred to the pipe by radiation is \(\begin{array}{lllll}\text { (a) } 0.24 & \text { (b) } 0.30 & \text { (c) } 0.37 & \text { (d) } 0.48 & \text { (e) } 0.58\end{array}\) (For air, use \(k=0.02401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.735, v=\) \(1.382 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) )

Consider three similar double-pane windows with air gap widths of 5,10 , and \(20 \mathrm{~mm}\). For which case will the heat transfer through the window will be a minimum?

A 0.1-W small cylindrical resistor mounted on a lower part of a vertical circuit board is \(0.3\) in long and has a diameter of \(0.2 \mathrm{in}\). The view of the resistor is largely blocked by another circuit board facing it, and the heat transfer through the connecting wires is negligible. The air is free to flow through the large parallel flow passages between the boards as a result of natural convection currents. If the air temperature at the vicinity of the resistor is \(120^{\circ} \mathrm{F}\), determine the approximate surface temperature of the resistor. Evaluate air properties at a film temperature of \(170^{\circ} \mathrm{F}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption? Answer: \(211^{\circ} \mathrm{F}\)

Why are the windows considered in three regions when analyzing heat transfer through them? Name those regions and explain how the overall \(U\)-value of the window is determined when the heat transfer coefficients for all three regions are known.
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