The burner on an electric stove has a power output of 2.0 kW. A 750 g stainless steel teakettle is filled with 20°C water and placed on the already hot burner. If it takes 3.0 min for the water to reach a boil, what volume of water, in cm3, was in the kettle? Stainless steel is mostly iron, so you can assume its specific heat is that of iron.
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20. Heat and Temperature
Specific Heat & Temperature Changes
Problem 34
Textbook Question
In an internal combustion engine, air at atmospheric pressure and a temperature of about 20°C is compressed in the cylinder by a piston to 1/9 of its original volume (compression ratio = 9.0). Estimate the temperature of the compressed air, assuming the pressure reaches 40 atm.

1
Start by identifying the given values: initial pressure \( P_1 = 1 \, \text{atm} \), initial temperature \( T_1 = 20^\circ \text{C} = 293 \, \text{K} \) (convert to Kelvin by adding 273), final pressure \( P_2 = 40 \, \text{atm} \), and the compression ratio \( V_1 / V_2 = 9.0 \).
Recognize that this is an adiabatic process (no heat exchange). For an adiabatic process, the relationship between pressure, volume, and temperature is governed by the equation \( P_1 V_1^\gamma = P_2 V_2^\gamma \), where \( \gamma \) is the adiabatic index (ratio of specific heats, \( C_p / C_v \)). For air, \( \gamma \approx 1.4 \).
Use the ideal gas law to relate temperature and pressure during the adiabatic process. The formula is \( \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{(\gamma - 1)/\gamma} \). Substitute the known values: \( T_1 = 293 \, \text{K} \), \( P_2 = 40 \, \text{atm} \), \( P_1 = 1 \, \text{atm} \), and \( \gamma = 1.4 \).
Simplify the exponent \( (\gamma - 1)/\gamma \): \( (1.4 - 1)/1.4 = 0.4/1.4 \approx 0.286 \). Substitute this into the equation: \( T_2 = T_1 \cdot \left( \frac{P_2}{P_1} \right)^{0.286} \).
Finally, calculate \( T_2 \) by substituting the values into the equation. This will give the temperature of the compressed air in Kelvin. If needed, convert back to Celsius by subtracting 273 from the result.

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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 context, it helps to understand how the compression of air affects its temperature and pressure, assuming the gas behaves ideally.
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Adiabatic Process
An adiabatic process is one in which no heat is exchanged with the surroundings. In an internal combustion engine, the rapid compression of air can be approximated as adiabatic, meaning that the temperature increase during compression can be calculated using specific heat ratios.
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Compression Ratio
The compression ratio is the ratio of the volume of the cylinder when the piston is at the bottom of its stroke to the volume when it is at the top. A higher compression ratio, such as 9.0 in this case, indicates a significant reduction in volume, which leads to an increase in temperature and pressure of the gas according to thermodynamic principles.
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Finding the Compression Ratio
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