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Chapter 15: Problem 6

A monatomic ideal gas at \(27^{\circ} \mathrm{C}\) undergoes a constant volume process from \(A\) to \(B\) and a constant pressure process from \(B\) to \(C\) Find the total work done during these two processes.

Short Answer

Expert verified
Answer: The total work done during the two processes is equal to the product of the constant pressure and the change in volume during the constant pressure process: \(W_{total} = P \Delta V\).

Step by step solution

01

Convert temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin. To do this, add 273.15 to the given Celsius temperature: \(T = 27^\circ \mathrm{C} + 273.15 = 300.15\,\mathrm{K}\)
02

Identify the equations for work done

During a constant volume process (from A to B), the work done is given by: \(W_{AB} = 0\) (since volume is constant) During a constant pressure process (from B to C), the work done is given by: \(W_{BC} = P \Delta V\) Total work done will be the sum of work done in each process: \(W_{total} = W_{AB} + W_{BC}\)
03

Find the constant pressure process work done

To find the work done in the constant pressure process, we will need more information about the pressure or volume change during the process. Since this is not provided in the exercise, we cannot calculate the work done for the constant pressure process as a numerical value. However, the total work done can still be expressed in terms of pressure and volume change: \(W_{total} = 0 + P \Delta V = P \Delta V\) So, the total work done during the two processes is equal to the product of the constant pressure and the change in volume during the constant pressure process.

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Most popular questions from this chapter

A fish at a pressure of 1.1 atm has its swim bladder inflated to an initial volume of \(8.16 \mathrm{mL}\). If the fish starts swimming horizontally, its temperature increases from \(20.0^{\circ} \mathrm{C}\) to $22.0^{\circ} \mathrm{C}$ as a result of the exertion. (a) since the fish is still at the same pressure, how much work is done by the air in the swim bladder? [Hint: First find the new volume from the temperature change. \(]\) (b) How much heat is gained by the air in the swim bladder? Assume air to be a diatomic ideal gas. (c) If this quantity of heat is lost by the fish, by how much will its temperature decrease? The fish has a mass of \(5.00 \mathrm{g}\) and its specific heat is about $3.5 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right)$.
A reversible heat engine has an efficiency of \(33.3 \%\) removing heat from a hot reservoir and rejecting heat to a cold reservoir at $0^{\circ} \mathrm{C} .$ If the engine now operates in reverse, how long would it take to freeze \(1.0 \mathrm{kg}\) of water at \(0^{\circ} \mathrm{C},\) if it operates on a power of \(186 \mathrm{W} ?\)
For a more realistic estimate of the maximum coefficient of performance of a heat pump, assume that a heat pump takes in heat from outdoors at $10^{\circ} \mathrm{C}$ below the ambient outdoor temperature, to account for the temperature difference across its heat exchanger. Similarly, assume that the output must be \(10^{\circ} \mathrm{C}\) hotter than the house (which itself might be kept at \(20^{\circ} \mathrm{C}\) ) to make the heat flow into the house. Make a graph of the coefficient of performance of a reversible heat pump under these conditions as a function of outdoor temperature (from $\left.-15^{\circ} \mathrm{C} \text { to }+15^{\circ} \mathrm{C} \text { in } 5^{\circ} \mathrm{C} \text { increments }\right)$
A heat pump delivers heat at a rate of \(7.81 \mathrm{kW}\) for $10.0 \mathrm{h} .\( If its coefficient of performance is \)6.85,$ how much heat is taken from the cold reservoir during that time?
A town is planning on using the water flowing through a river at a rate of \(5.0 \times 10^{6} \mathrm{kg} / \mathrm{s}\) to carry away the heat from a new power plant. Environmental studies indicate that the temperature of the river should only increase by \(0.50^{\circ} \mathrm{C} .\) The maximum design efficiency for this plant is \(30.0 \% .\) What is the maximum possible power this plant can produce?
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