∂ P ∂ T V = − ( ∂ V / ∂ T ) P ( ∂ V / ∂ P ) T For the second expression you should obtain Then express it more abstractly in terms of the partial derivatives used to define β and κ T. Express this partial derivative in terms of β and κ T. Then the ratio of d P to d T is equal to ( ∂ P / ∂ T ) V, since there is no net change in volume. (c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Write the change in volume for this process, d V 2, in terms of d P and the isothermal compressibility κ T, defined as (b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume, d V 1, in terms of d T and the thermal expansion coefficient β introduced in Problem 1.7. (a) First imagine slightly increasing the temperature of a material at constant pressure. To see why, estimate the pressure needed to keep V fixed as T increases, as follows. Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. substitute with k = 1.38 × 10 − 23 m 2 ⋅ kg ⋅ s − 2 ⋅ K − 1, at atmospheric pressure P = 1 atm = 101325 Pa, and at room temperature T = 300 ∘ K Where r is the effective radius of a helium atom, r = 1.4 × 10 − 10 m. The mean free path ℓ is based on the idea that the length of a cylinder with a radius equal to the molecule's diameter and volume equal to the average volume per molecule is equal to the length of a cylinder with a radius equal to the molecule's diameter and volume equal to the average volume per molecule V N, so that: Substitute k = 1.38 × 10 − 23 m 2 ⋅ kg ⋅ s − 2 ⋅ K − 1, and at room temperature T = 300 ∘ K, and m is the mass of helium which is about 4 atomic mass units or m = 4 × 1.66 × 10 − 27 = 6.64 × 10 − 27 kg, so the average molecular velocity is therefore: Where localid="1650283569119" v ¯ is the average molecular velocity, from which we can find the approximate using RMS speed, which is: The approximation formula can be used to calculate the thermal conductivity of a gas such as helium.
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