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Lesson_1.4_Printable_PPT

# Lesson_1.4_Printable_PPT - Heat Capacities of Gases Heat...

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Heat Capacities of Gases. Heat capacity of a substance is a measure of the amount of heat needed to raise the temperature of the substance by 1 K. Heat capacity represented by C is: If a quantity of heat dQ added to a C = mass × specific heat capacity = mc. dQ = C dT substance increases the temperature of the substance by dT , then dQ C dT =

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The amount of heat needed to raise the temperature of a given mass of gas by I K depends on whether the gas is kept at constant volume or constant pressure. Therefore, we have to define two types of heat capacities for a gas. Heat capacity of a gas at constant pressure is represented by C p Heat capacity of a gas at constant volume is represented by C v
1. Heat Capacity at constant volume, C v . When a gas is kept at constant volume, all the heat added to it ( dQ ) goes into increasing its internal energy ( dU ) as the gas does no work in expanding. dQ = dU dQ dU dT dT = But dQ C dT = v dU C dT =

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2. Heat Capacity at Constant Pressure, C p . When a gas is heated at constant pressure, it expands and does work in addition to increasing its internal energy. dQ = dU + dW Since v dU C dT = dW = P dV v dU = C dT dQ = C v dT + PdV p dQ C dT = v dQ dV C P dT dT = + Divide by dT p v dV C C P dT = +
We now find dV/dT using the relation: P V = nR T Take the differential of both sides. p v C C dT P dV = + PdV = nR dT dV P nR dT = p v C C nR = + C p = C v + nR

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What is internal energy U of n moles of a gas? 3 2 U nRT = But v dU dT C = 3 2 dU nR dT = 3 C nR = Take derivative of U with respect to T 3 2 n nR R = + 5 2 p C nR = 2 v C p = C v + nR
Molar Heat Capacity Molar heat capacity is the heat capacity per mole and we use C mp and C mv to represent molar heat capacities at constant pressure and constant volume respectively. Molar heat capacity is measured in J mol -1 K -1 p mp C C n = 5 2 p C nR = 3 2 v C nR = 5 2 R = v mv C C n = 3 2 R = C – C = R This means that the molar heat capacities have constant values for all the gases such that C mp = 2.5 R = 20.79 J.mol -1 .K -1 C mv = 1.5 R = 12.47 J.mol -1 .K -1 mp mv Measured values of C mp and C mv shows that for monatomic gases, this prediction is true while it fails for diatomic and polyatomic gases.

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Gas C mp C mv C mp – C mv Monatomic He Ne Ar Kr Xe 20.79 20.79 20.79 20.79 20.79 12.52 12.68 12.45 12.45 12.52 8.27 8.11 8.34 8.34 8.27 Diatomic 29.12 20.80 8.32 C mp = 20.79 J.mol -1 .K -1 C mv = 12.47 J.mol -1 .K -1 N 2 H 2 O 2 28.82 29.37 20.44 20.98 8.38 8.39 Polyatomic CO 2 N 2 O 36.62 36.90 28.17 28.39 8.45 8.51 The table shows the experimental values of C mp and C mv for common gases.
Although experimental values for diatomic and polyatomic gases differ from the predicted values, the values of C mp – C mv for all gases agree perfectly with prediction. We will explain this discrepancy using the equipartition theorem . In the lesson on Kinetic Theory of Gases, you saw that the energy associated with motion of a molecule in the x direction is ½ kT The ability of a molecule to move in the x, y and z directions are called degrees of freedom.

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