Chapter 1: Q.1.43 (page 31)

Calculate the heat capacity of liquid water per molecule, in terms of K . Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have?

Short Answer

Expert verified

Short Answer :

Heat capacity per molecule $C_{m} = 9.072 k$

Degrees of freedom f = 18

Step by step solution

Given Information :

Water's thermal energy is stored in quadratic degrees of freedom

Explanation

The specific heat capacity of water is $c_{water} = 1 cal \cdot g^{- 1} \cdot C^{- 1} = 4.186 J^{- 1} g^{- 1} \cdot C^{- 1}$ , and the molar weight of water is $18.015 g \cdot$ mol $^{- 1}$ so in $1 gram$ of water.

$N = \frac{Number of molecules in one mole}{Mass of one mole} = \frac{6.022 \times 10^{23}}{18.015}$

$N = 3.343 \times 10^{22}$ molecules

The heat capacity of a single molecule is therefore:

$C_{m} = \frac{C}{N} = \frac{4.186}{3.343 \times 10^{22}} = 1.252 \times 10^{- 22} J \cdot C^{- 1} C_{m} = 1.252 \times 10^{- 22} \times \frac{k}{k} = 1.252 \times 10^{- 22} \times \frac{k}{1.38 \times 10^{- 23}} C_{m} = 9.072 k$

Explanation

The heat capacity is linearly proportional to the number of degrees of freedom, according to the equipartition theorem,

$C_{m} = f \frac{k}{2}$

Therefore now on comparing heat capacities, we have

$f \frac{k}{2} = 9.072 k$

$f = 18$

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Calculate the heat capacity of liquid water per molecule, in terms of K . Suppose (incorrectly) that all the thermal energy of water is stored in quadratic degrees of freedom. How many degrees of freedom would each molecule have to have?

Short Answer

Step by step solution

Given Information :

Explanation

Explanation

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