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Note: The important things to remember from these heat capacity derivations are the final expressions for the two heat capacities as partial derivatives, i.e. $\\mathrm{C_v=(\\frac{\\delta U}{\\delta T})_v}$ and $\\mathrm{C_p=(\\frac{\\delta H}{\\delta T})_p}$, and what these expressions mean in reality.

The heat capacity of a system is mathematically defined as the ratio of a minimal heat transfer to its corresponding infinitesimal (differential) temperature change in the system, i.e. the derivative of heat transfer with respect to temperature:

$\\mathrm{C=\\frac{dQ}{dT}}$ .

More specifically, the heat capacity at constant volume, $\\mathrm{C_v}$, and the heat capacity at constant pressure, $\\mathrm{C_p}$, are defined as the partial derivatives of heat transfer with respect to temperature at constant volume and pressure, respectively:

$\\mathrm{C_v=(\\frac{\\delta Q}{\\delta T})_v}$ ,

$\\mathrm{C_p=(\\frac{\\delta Q}{\\delta T})_p}$ .

Heat transfer, $\\mathrm{Q}$, is not a state function - to instead express our heat capacities in terms of state functions, e.g. internal energy, we must utilise differential expressions for heat transfer.

For the heat capacity at constant volume, we use the differential formulation of the first law of thermodynamics:

$\\mathrm{dU=\\delta Q+p\\space dV}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\delta Q=dU+p\\space dV}$ .

We then split our $\\mathrm{dU}$ term into an expression for $\\mathrm{dU}$ in terms of its temperature and volume partial derivatives (known as differentiating by parts), i.e. $\\mathrm{dU=(\\frac{\\delta U}{\\delta T})_v\\space dT+(\\frac{\\delta U}{\\delta V})_T\\space dV}$, giving us:

$\\mathrm{\\delta Q=(\\frac{\\delta U}{\\delta T})_v\\space dT+(\\frac{\\delta U}{\\delta V})_T\\space dV+p\\space dV}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\delta Q=(\\frac{\\delta U}{\\delta T})_v\\space dT+[(\\frac{\\delta U}{\\delta V})_T\\space +p]dV}$ .

For constant volume, we will set $\\mathrm{dV=0}$, completely nullifying our term with the square brackets:

$\\mathrm{(\\delta Q)_v=(\\frac{\\delta U}{\\delta T})_v\\space dT}$ .

Finally, we then effectively divide through by the temperature differential, $\\mathrm{dT}$, to yield our heat capacity at constant volume:

$\\mathrm{C_v=(\\frac{\\delta Q}{\\delta T})_v=(\\frac{\\delta U}{\\delta T})_v}$ .

This means that heat capacity at constant volume is defined as the slope of internal energy with respect to temperature (at constant volume).

Now, for the heat capacity at constant pressure, we know we will be able to apply $\\mathrm{dp=0}$, so we want to create an expression where this will be useful - we begin by using our earlier trick, differentiation by parts, to express the differentials of both internal energy and volume each in terms of their temperature and pressure partial derivatives:

$\\mathrm{dU=(\\frac{\\delta U}{\\delta T})_p\\space dT+(\\frac{\\delta U}{\\delta p})_T\\space dp}$ ,

$\\mathrm{dV=(\\frac{\\delta V}{\\delta T})_p\\space dT+(\\frac{\\delta V}{\\delta p})_T\\space dp}$ .

We then substitute these into our differential formulation of the first law of thermodynamics:

$\\mathrm{\\delta Q=dU+p\\space dV}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\delta Q=(\\frac{\\delta U}{\\delta T})_p\\space dT+(\\frac{\\delta U}{\\delta p})_T\\space dp+p[(\\frac{\\delta V}{\\delta T})_p\\space dT+(\\frac{\\delta V}{\\delta p})_T\\space dp]}$ .

We can then put our fact (that for constant pressure we have $\\mathrm{dp=0}$) to use, nullifying all terms which are multiplied by $\\mathrm{dp}$ and leaving us with:

$\\mathrm{(\\delta Q)_p=(\\frac{\\delta U}{\\delta T})_p\\space dT+p(\\frac{\\delta V}{\\delta T})_p\\space dT}$ .

We then effectively divide through by the temperature differential, $\\mathrm{dT}$, to yield our heat capacity at constant pressure:

$\\mathrm{C_p=(\\frac{\\delta Q}{\\delta T})_p=(\\frac{\\delta U}{\\delta T})_p+p(\\frac{\\delta V}{\\delta T})_p}$ .

Finally, we note that this can be re-expressed as a single partial derivative of a summed quantity:

$\\mathrm{C_p=(\\frac{\\delta Q}{\\delta T})_p=\\frac{\\delta}{\\delta T})_p(U+pV)}$ .

This sum, $\\mathrm{U+pV}$, we will denote as a new state function called enthalpy, $\\mathrm{H}$. Thus, we have:

$\\mathrm{C_p=(\\frac{\\delta Q}{\\delta T})_p=(\\frac{\\delta H}{\\delta T})_p}$ .

This means that heat capacity at constant pressure is defined as the slope of enthalpy with respect to temperature (at constant pressure).

The entropy change resulting from a change in temperature of a gas is given by the equation:

$\\mathrm{\\Delta S=nCln(\\frac{T_2}{T_1})}$ ,

where $\\mathrm{n}$ is the number of moles of the gas; $\\mathrm{C}$ is the specific heat capacity of the material in question (under the conditions specified); $\\mathrm{T_1}$ is the initial temperature of the gas; and $\\mathrm{T_2}$ is the final temperature of the gas.

We are given an initial temperature of $\\mathrm{T_1=\\var{ctemp1} \\space ^\\circ C}$, and a final temperature of $\\mathrm{T_2=\\var{ctemp2} \\space ^\\circ C}$, and asked to calculate the resulting change in entropy for the carbon dioxide.

We must remember to convert both of our temperatures to Kelvin for use in the equation, by adding $\\mathrm{273.15}$:

$\\mathrm{T_1=\\var{ctemp1} \\space ^\\circ C=\\var{ktemp1}\\space K}$ ,

$\\mathrm{T_2=\\var{ctemp2} \\space ^\\circ C=\\var{ktemp2}\\space K}$ .

We are also given the mass of the carbon dioxide gas, $\\mathrm{m=\\var{mass}\\space g}$ - to use our equation, we must first obtain the number of moles of carbon dioxide gas, which we can calculate from the mass:

$\\mathrm{n=\\frac{m}{gFM}=\\frac{\\var{mass}}{\\var{gfm}}=(\\var{sigformat(mol,5)}\\cdots)\\space mol}$ .

Additionally, we are provided the molar heat capacities for carbon dioxide at constant volume, $\\mathrm{C_v=28.46\\space J\\space mol^{-1}\\space K^{-1}}$, and at constant pressure, $\\mathrm{C_p=36.94\\space J\\space mol^{-1}\\space K^{-1}}$ . Since our system is in a sealed, rigid vessel, it cannot expand during heating, and so its volume will remain constant throughout - thus, we will use $\\mathrm{C_v=28.46\\space J\\space mol^{-1}\\space K^{-1}}$ in our equation to calculate the entropy change.

We can now subsitute our values into our original equation to obtain the entropy change:

$\\mathrm{\\Delta S=nCln(\\frac{T_2}{T_1})}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\Delta S=nC_vln(\\frac{T_2}{T_1})}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\Delta S=(\\var{sigformat(mol,5)}\\cdots)\\times28.46 \\times ln(\\frac{\\var{ktemp2}}{\\var{ktemp1}})=\\var{sigformat(entro,3)}\\space J \\space K^{-1}}$ .

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Complete the following definitions:

Heat capacity at constant volume, $\\mathrm{C_v}$ , is mathematically defined as the slope of [[1]] with respect to temperature.

Heat capacity at constant pressure, $\\mathrm{C_p}$ , is mathematically defined as the slope of [[2]] with respect to temperature.

For carbon dioxide, $\\mathrm{C_p=36.94\\space J\\space mol^{-1}\\space K^{-1}}$ and $\\mathrm{C_v=28.46\\space J\\space mol^{-1}\\space K^{-1}}$.

Calculate the change in entropy of the system when $\\mathrm{\\var{mass}\\space g}$ of carbon dioxide gas in a sealed, rigid vessel is heated from $\\mathrm{\\var{ctemp1}^\\circ C}$ to $\\mathrm{\\var{ctemp2}^\\circ C}$ :

[[0]] $\\mathrm{J\\space K^{-1}}$ .

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