An experimenter adds 970970 J of heat to 1.751.75 mol of an ideal gas to heat it from 10.010.0°C to 25.025.0°C at constant pressure. The gas does +223+223 J of work during the expansion. Calculate γ\gamma for 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

Problem 23b

Chapter 19, Problem 23b

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.

Verified step by step guidance

First, understand that the heat added to the system (Q) is 970 J, and the work done by the gas (W) is 223 J. The change in internal energy (ΔU) can be found using the first law of thermodynamics: ΔU = Q - W.

Calculate the change in internal energy (ΔU) using the values provided: ΔU = 970 J - 223 J.

Next, recall that for an ideal gas, the change in internal energy (ΔU) is also related to the change in temperature (ΔT) and the molar heat capacity at constant volume (C_v) by the equation: ΔU = n * C_v * ΔT, where n is the number of moles.

Determine the change in temperature (ΔT) in Kelvin. Since the temperature change is from 10.0°C to 25.0°C, ΔT = 25.0°C - 10.0°C = 15.0°C. Convert this to Kelvin, noting that the change in Celsius is the same as the change in Kelvin.

Finally, use the relationship between the heat capacities at constant pressure (C_p) and constant volume (C_v) to find the heat capacity ratio (γ), where γ = C_p / C_v. Recall that C_p = C_v + R, where R is the ideal gas constant (8.314 J/mol·K). Use the calculated ΔU and ΔT to find C_v, then determine γ.

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Key Concepts

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

First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system. Mathematically, it is expressed as ΔU = Q - W, where ΔU is the change in internal energy, Q is the heat added, and W is the work done by the system. This principle is crucial for understanding energy conservation in thermodynamic processes.

Specific Heat Capacity at Constant Pressure (Cp)

Specific heat capacity at constant pressure, Cp, is the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius at constant pressure. For an ideal gas, Cp is related to the change in enthalpy and is used to calculate the heat added during a process at constant pressure. It is essential for determining how much energy is needed to change the temperature of a gas under these conditions.

Heat Capacity Ratio (γ)

The heat capacity ratio, γ (gamma), is the ratio of the specific heat capacity at constant pressure (Cp) to the specific heat capacity at constant volume (Cv), expressed as γ = Cp/Cv. This ratio is significant in thermodynamics as it influences the behavior of gases during adiabatic processes and is used to calculate various thermodynamic properties, such as the speed of sound in a gas and the efficiency of heat engines.

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An experimenter adds 970970 J of heat to 1.751.75 mol of an ideal gas to heat it from 10.010.0°C to 25.025.0°C at constant pressure. The gas does +223+223 J of work during the expansion. Calculate γ\(\gamma\) for the gas.

Verified video answer for a similar problem:

Key Concepts

First Law of Thermodynamics

Specific Heat Capacity at Constant Pressure (Cp)

Heat Capacity Ratio (γ)

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.