Determine the internal energy change \triangle u of hydrogen, in

Step 1:
(a) Using the empirical specific heat equation as a function of temperature (Table A-2c):
The empirical specific heat equation for hydrogen is given by:
$c_p=a+bT+c{T}^2+d{T}^3$
where $c_p$ is the specific heat capacity at constant pressure, $T$ is the temperature in Kelvin, and $a$, $b$, $c$, and $d$ are coefficients provided in Table A-2c.
To determine the internal energy change, we need to integrate the specific heat capacity expression with respect to temperature:
$\Delta u={\int }_{T_1}^{T_2}{c}_{p}dT$
where $T_1$ is the initial temperature (200 K) and $T_2$ is the final temperature (800 K).
Integrating the specific heat capacity equation and evaluating it between the temperature limits, we have:
$\Delta u={\int }_{200}^{800}(a+bT+c{T}^2+d{T}^3)dT$
Solving this integral, we obtain the expression for the internal energy change in terms of the coefficients $a$, $b$, $c$, and $d$.
Step 2:
(b) Using the $c_v$ value at the average temperature (Table A-2b):
The $c_v$ value for hydrogen at the average temperature can be obtained from Table A-2b. Let's denote this value as $c_v^{avg}$.
The internal energy change can be calculated using the relationship:
$\Delta u=c_v^{avg}·(T_2-T_1)$
where $T_1$ is the initial temperature (200 K) and $T_2$ is the final temperature (800 K).
Step 3:
(c) Using the $c_v$ value at room temperature (Table A-2a):
Similarly, the $c_v$ value for hydrogen at room temperature can be obtained from Table A-2a. Let's denote this value as $c_v^{room}$.
The internal energy change can be calculated using the relationship:
$\Delta u=c_v^{room}·(T_2-T_1)$
where $T_1$ is the initial temperature (200 K) and $T_2$ is the final temperature (800 K).

To determine the internal energy change $\Delta u$ for hydrogen as it is heated from 200 to 800 K, we can use three different approaches:
(a) The empirical specific heat equation as a function of temperature (Table A-2c):
The empirical specific heat equation for hydrogen is given by:
$c_p=a+bT+c{T}^2+d{T}^3+e{T}^4$ where $c_p$ is the specific heat at constant pressure, $T$ is the temperature in Kelvin, and $a$, $b$, $c$, $d$, and $e$ are constants.
To calculate the internal energy change, we can integrate the specific heat equation with respect to temperature:
$\Delta u={\int }_{T_1}^{T_2}{c}_{p}dT$
Substituting the values of $T_1=200$ K and $T_2=800$ K into the equation, we have:
$\Delta u={\int }_{200}^{800}(a+bT+c{T}^2+d{T}^3+e{T}^4)dT$
(b) The $c_v$ value at the average temperature (Table A-2b):
The specific heat at constant volume, $c_v$, is given as a constant value at different temperatures in Table A-2b. We can use the average value of $c_v$ over the temperature range to calculate the internal energy change.
(c) The $c_v$ value at room temperature (Table A-2a):
Similar to approach (b), we can use the $c_v$ value at room temperature from Table A-2a to calculate the internal energy change.
Let's proceed with solving these approaches one by one.
(a) Using the empirical specific heat equation:
The constants $a$, $b$, $c$, $d$, and $e$ for hydrogen can be obtained from Table A-2c. We can substitute these values into the equation and integrate over the temperature range to find the internal energy change.
(b) Using the $c_v$ value at the average temperature:
From Table A-2b, we can find the $c_v$ value at the average temperature, which is the average of the initial and final temperatures (200 K and 800 K in this case). We can multiply this value by the temperature difference to calculate the internal energy change:
$\Delta u=c_v·\Delta T$
(c) Using the $c_v$ value at room temperature:
From Table A-2a, we can find the $c_v$ value at room temperature (which is usually around 298 K). We can multiply this value by the temperature difference to calculate the internal energy change:
$\Delta u=c_v·\Delta T$
Now you have the three different approaches to calculate the internal energy change for hydrogen as it is heated from 200 to 800 K.