A monatomic ideal gas that is initially at 1.50×1051.50\$$times10^51.50×105 Pa $$and has a volume of 0.08000.08000.0800 m3 is compressed adiabatically to a volume of 0.04000.04000.0400 m3. What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 19: The First Law of Thermodynamics

Problem 27c

Chapter 19, Problem 27c

A monatomic ideal gas that is initially at $1.50$\times$10^5$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the ratio of the final temperature of the gas to its initial temperature? Is the gas heated or cooled by this compression?

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First, understand that for an adiabatic process involving an ideal gas, the relationship between pressure, volume, and temperature is given by the adiabatic condition: $ PV^\gamma = \text{constant} $, where $ \gamma $ (gamma) is the heat capacity ratio $ C_p/C_v $. For a monatomic ideal gas, $ \gamma = \frac{5}{3} $.

Next, use the adiabatic condition to relate the initial and final states of the gas. Since $ PV^\gamma = \text{constant} $, we can write: $ P_1 V_1^\gamma = P_2 V_2^\gamma $.

To find the ratio of the final temperature $ T_2 $ to the initial temperature $ T_1 $, use the relation derived from the ideal gas law: $ \frac{T_2}{T_1} = \left(\frac{V_1}{V_2}\right)^{\gamma - 1} $.

Substitute the given volumes into the equation: $ \frac{T_2}{T_1} = \left(\frac{0.0800}{0.0400}\right)^{\frac{5}{3} - 1} $.

Finally, determine whether the gas is heated or cooled. In an adiabatic compression, the work done on the gas increases its internal energy, which increases the temperature. Therefore, the gas is heated during this compression.

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Key Concepts

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

Adiabatic Process

An adiabatic process is a thermodynamic process in which no heat is exchanged with the surroundings. For an ideal gas, this means that any change in internal energy is due solely to work done on or by the system. In an adiabatic compression, the gas is compressed without heat loss, leading to an increase in temperature.

Ideal Gas Law

The ideal gas law is a fundamental equation that relates the pressure, volume, and temperature of an ideal gas through the equation PV = nRT, where P is pressure, V is volume, T is temperature, n is the number of moles, and R is the ideal gas constant. This law helps in understanding how changes in one property affect the others in a gas system.

Adiabatic Equation for Ideal Gases

For an adiabatic process involving an ideal gas, the relationship between pressure, volume, and temperature is given by the equation PV^γ = constant, where γ (gamma) is the heat capacity ratio (Cp/Cv). This equation can be used to find the ratio of final to initial temperatures using the relation T2/T1 = (V1/V2)^(γ-1), indicating how temperature changes with volume during adiabatic compression.

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Textbook Question

A monatomic ideal gas that is initially at $1.50 \times 10^{5}$ Pa and has a volume of $0.0800$ m³ is compressed adiabatically to a volume of $0.0400$ m³. What is the final pressure?

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Textbook Question

On a warm summer day, a large mass of air (atmospheric pressure $1.01 \times 10^{5}$ Pa) is heated by the ground to $26.0$ °C and then begins to rise through the cooler surrounding air. (This can be treated approximately as an adiabatic process; why?) Calculate the temperature of the air mass when it has risen to a level at which atmospheric pressure is only $0.850 \times 10^{5}$ Pa. Assume that air is an ideal gas, with $g equals 1.40$ . (This rate of cooling for dry, rising air, corresponding to roughly $1$ C° per $100$ m of altitude, is called the dry adiabatic lapse rate.)

Textbook Question

An experimenter adds $970$ J of heat to $1.75$ mol of an ideal gas to heat it from $10.0$ °C to $25.0$ °C at constant pressure. The gas does $positive 223$ J of work during the expansion. Calculate $γ$ for the gas.

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