General guidance

Concepts and reason

Internal energy is a state function. For a ideal gas, it depends on only temperature irrespective of pressure and volume.

If the tabulated data obtained from the experimental analysis for a particular gas is available, then the internal energy change of gas is calculated using available data from data tables. Some times, it is also calculated by obtaining average specific heat ${c_{v,avg}}$ at average temperature and then multiplying ${c_{v,avg}}$ with temperature difference.

Calculation of change in internal energy is generally used in thermal engineering applications.

Fundamentals

Internal energy:

The energy associated with random motion of molecules at microscopic level is called internal energy.

The internal energy of the ideal gas is function of temperature only.

$u = u(T)$

The differential change of internal energy can be expressed as,

$du = {c_v}(T)\,\,dT$

The change in internal energy for an ideal gas during a process from state 1 to state 2 is,

$\Delta u = \int\limits_1^2 {{c_v}(T)\,\,dT} \,$

Observe that the equations of integration are applicable to all processes. It does not mean that the term ${c_v}$ in the equation relates to a constant volume process. Also, the term ${c_p}$ does not apply only to a constant pressure process.

There are many ways to determine the internal energy of ideal gases.

• Using the tabulated values of u, when the data tables are available.

• Using integration method and ${c_v}$ as a function of temperature. This is a difficult procedure but, accurate results can be obtained.

• Using average specific heats, it is easy to calculate the internal energy. The average specific heats are calculated from the average of temperatures, that is $\frac{{{T_1} + {T_2}}}{2}$ . if the final temperature is not known, specific heats are calculated at an anticipated average temperature. Then, ${T_2}$ is calculated from average specific heats. ${T_2}$ can be redefined, if required, by considering a specific heat at a new average temperature.

The equation for change in internal energy is,

${u_2} - {u_1} = {c_{v,avg}}({T_2} - {T_1})$

Here, ${T_2}$ is the final temperature, ${T_1}$ is the initial temperature, and ${c_{v,avg}}$ is the average specific heat at constant volume.

Conversion of units of internal energy:

$\Delta u = \frac{{\Delta \bar u}}{M}$

Here, $\Delta \bar u$ is internal energy interms of ${\rm{kJ/Kmol}}$ ,M is the molecular mass of the gas interms of ${\rm{kg/Kmol}}$ , and $\Delta u$ is the internal energy interms of ${\rm{kJ/kg}}$ .

First Step | All Steps | Answer Only

Step-by-step

Step 1 of 4

a)

Obtain the empirical equation for specific heat at constant pressure, from TABLE A-2c “ideal-gas specific heats of various common gases”.

${\bar c_p}(T) = a + bT + c{T^2} + d{T^3}$

For hydrogen,

$\begin{array}{l}\\a{\rm{ = }}29.11\\\\{\rm{b}} = - 0.{\rm{1916}} \times {\rm{1}}{0^{ - {\rm{2}}}}\\\\{\rm{c}} = 0.{\rm{4}}00{\rm{3}} \times {\rm{1}}{0^{ - {\rm{5}}}}\\\\{\rm{d}} = - 0.{\rm{87}}0{\rm{4}} \times {\rm{1}}{0^{ - {\rm{9}}}}\\\end{array}$

Write the expression for calculating the specific heat constant volume ${\bar c_v}\left( T \right)$ .

$\begin{array}{l}\\{{\bar c}_p}\left( T \right) - {{\bar c}_v}\left( T \right) = {R_u}\,\,\,\,\,\left( {{\rm{kJ/kmol}} \cdot {\rm{K}}} \right)\\\\{{\bar c}_v}\left( T \right) = {{\bar c}_p}\left( T \right) - {R_u}\\\end{array}$

Here, ${R_u}$ is universal gas constant.

Substitute $a + bT + c{T^2} + d{T^3}$ for ${\bar c_p}(T)$ .

${\bar c_v}\left( T \right) = a + bT + c{T^2} + d{T^3} - {R_u}$

Calculate the internal energy as follows:

$\Delta \bar u = \int\limits_{{T_1}}^{{T_2}} {{{\bar c}_v}(T)\,\,dT} \,$

Substitute $a + bT + c{T^2} + d{T^3} - {R_u}$ for ${\bar c_v}\left( T \right)$ .

$\begin{array}{l}\\\Delta \bar u = \int\limits_{{T_1}}^{{T_2}} {(a - {R_u} + bT + c{T^2} + d{T^3})} {\rm{dT}}\\\\\Delta \bar u = \left( {a - {R_u}} \right)\left( T \right)_{{T_1}}^{{T_2}}{\rm{ + b}}\left( {\frac{{{T^2}}}{2}} \right)_{{T_1}}^{{T_2}} + c\left( {\frac{{{T^3}}}{3}} \right)_{{T_1}}^{{T_2}} + d\left( {\frac{{{T^4}}}{4}} \right)_{{T_1}}^{{T_2}}\\\\\Delta \bar u = \left( {a - {R_u}} \right)\left( {{T_2} - {T_1}} \right){\rm{ + 1/2b}}\left( {T_2^2 - {T_1}^2} \right){\rm{ + 1/3c}}\left( {T_2^3 - {T_1}^3} \right){\rm{ + 1/4d}}\left( {T_2^4 - {T_1}^4} \right)\\\end{array}$

Determine the internal energy change of hydrogen, in kJ/kg, as it is heatedfrom 200 to 800 K, using (a) the empirical specific heat equation as a function oftemperature (Table A 2c), (b) the cv value at the average temperature (Table A-2b), and(c) the cv value at room temperature (Table A 2a).

Public Answer

Determine the internal energy change of hydrogen, in kJ/kg, as it is heated
from 200 to 800 K, using (a) the empirical specific heat equation as a function of
temperature (Table A 2c), (b) the cv value at the average temperature (Table A-2b), and
(c) the cv value at room temperature (Table A 2a).