joule showed using this experimental apparatus that u=u(T)

Figure 4-22 Schematic of the experimental apparatus used by Joule.

Internal energy and enthalpy change of an ideal gas

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For ideal gases, u, h, cv, and cp vary with temperature only.

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At low pressures, all real gases approach ideal-gas behavior, and therefore their specific heats depend on temperature only.
The specific heats of real gases at low pressures are called ideal-gas specific heats, or zero-pressure specific heats, and are often denoted c_p0 and c_v0.
u and h data for a number of gases have been tabulated.
These tables are obtained by choosing an arbitrary reference point and performing the integrations by treating state 1 as the reference state.

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Ideal-gas constant-pressure specific heats for some gases (see Table A–2c for cp equations).

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Ideal-gas constant-pressure specific heats for some gases (see Table A–2c for cp equations).

Internal energy and enthalpy change when specific heat is taken constant at an average value

1.316-2.png $(\frac{k J}{k g})$

Figure 4-27 The relation $Δ$ u = c_{v $Δ$}_{T is valid for any kind of process, constant-volume or not.}

Figure 4-26 For small temperature intervals, the specific heats may be assumed to vary linearly with temperature.

Three ways of calculating $Δ$ u and $Δ$ h

Three ways of calculating $Δ$ u

Specific Heat Relations of Ideal Gases

The relationship between cp, cv and R

On a molar basis

Specific heat ratio

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The specific ratio varies with temperature, but this variation is very mild.
For monatomic gases (helium, argon, etc.), its value is essentially constant at 1.667.

Many diatomic gases, including air, have a specific heat ratio of about 1.4 at room temperature.

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Heating of a Gas in a Tank by Stirring

Heating of a Gas by a Resistance Heater

Heating of a Gas at Constant Pressure

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