Two moles of an ideal gas are heated at constant pressure from T=27T = 27°C to T=107T = 107°C. Calculate the work done by the gas.

Ch 19: The First Law of Thermodynamics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 19: The First Law of Thermodynamics

문제 1b

19장, 문제 1b

Two moles of an ideal gas are heated at constant pressure from $T equals 27$ °C to $T equals 107$ °C. Calculate the work done by the gas.

검증된 단계별 안내

First, convert the initial and final temperatures from Celsius to Kelvin by adding 273.15 to each temperature. This gives you the initial temperature $ T_1 = 300.15 \text{ K} $ and the final temperature $ T_2 = 380.15 \text{ K} $.

Recall that the work done by an ideal gas at constant pressure is given by the formula $ W = P \Delta V $, where $ \Delta V $ is the change in volume. However, we can also use the relation $ W = nR\Delta T $ for an ideal gas, where $ n $ is the number of moles, $ R $ is the ideal gas constant, and $ \Delta T $ is the change in temperature in Kelvin.

Calculate the change in temperature $ \Delta T = T_2 - T_1 $. Substitute the values to find $ \Delta T = 380.15 \text{ K} - 300.15 \text{ K} $.

Use the ideal gas constant $ R = 8.314 \text{ J/mol·K} $ and the number of moles $ n = 2 $ to calculate the work done: $ W = nR\Delta T $. Substitute the values into the equation.

Finally, simplify the expression to find the work done by the gas. Remember, the units of work will be in joules (J).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature in Kelvin. It describes the relationship between these variables for an ideal gas, allowing us to predict how a gas will behave under different conditions.

Work Done by a Gas at Constant Pressure

When a gas expands or contracts at constant pressure, the work done by the gas is given by the formula W = PΔV, where W is the work done, P is the constant pressure, and ΔV is the change in volume. This concept is crucial for calculating the energy transferred as work when a gas undergoes a thermodynamic process at constant pressure.

Temperature Conversion

In thermodynamics, temperature must be expressed in Kelvin for calculations involving the Ideal Gas Law. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. This conversion is essential for ensuring that temperature values are compatible with the equations used in gas law calculations.

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교과서 질문

The graph in Fig. E $19.4$ shows a $pV$ -diagram of the air in a human lung when a person is inhaling and then exhaling a deep breath. Such graphs, obtained in clinical practice, are normally somewhat curved, but we have modeled one as a set of straight lines of the same general shape. (Important: The pressure shown is the gauge pressure, not the absolute pressure.) The process illustrated here is somewhat different from those we have been studying, because the pressure change is due to changes in the amount of gas in the lung, not to temperature changes. (Think of your own breathing. Your lungs do not expand because they've gotten hot.) If the temperature of the air in the lung remains a reasonable $20$ °C, what is the maximum number of moles in this person's lung during a breath?

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교과서 질문

Two moles of an ideal gas are heated at constant pressure from $T = 27$ °C to $T = 107$ °C. Draw a $p V$ -diagram for this process.

Six moles of an ideal gas are in a cylinder fitted at one end with a movable piston. The initial temperature of the gas is $27.0$ °C and the pressure is constant. As part of a machine design project, calculate the final temperature of the gas after it has done $2.40 times 10 cubed$ J of work.

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