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

Gregory Emery

Gregory Emery

Answered question

2021-12-15

Determine the internal energy change u 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).

Answer & Explanation

vrangett

vrangett

Beginner2021-12-16Added 36 answers

Step 1
Given:
- Initial temperature T1=200K
- Final temperature T2=800K
Required
- Determine the internal energy change of hydrogen using
a) The empirical specific heat equation
b) The cv value at average temperature
c) The cv value at room temperature
Step 2
Solution
Part a
- Using the empirical relation of cp(T) from table (A-2C) and relating it to cv(T)
cv(T)=cpRu=(aRu)+bT+cT2+dT3
Where (a=29.11,b=0.1916×102,c=0.4003×105,d=0.8704×109)
- The internal energy change could be defined as the following
u=12cv(T)dT=12[(aRu)+bT+cT2+dT3]dT
u=(aRu)(T2T1)+12b(T22T12)+13c(T23T13)+14d(T24T14)
u=(29.118.314)(800200)+12×(0.1916×102)×(80022002)+13×(0.4003×105)×(80032003)+14×(0.8704×109)×(80042004)=12487KJKmol
Marcus Herman

Marcus Herman

Beginner2021-12-17Added 41 answers

Step 1
a) In this problem we need to determine the internal energy change u by using three different methods.
The first method is with the empirical specific heat equation. The equation gives us cp and we need the cv for the calculation.
cp=a+bT+cT2+dT3
cv=cpRu
cv=aRu+bT+cT2+dT3
The values for the constants a,b,c and d we find in the table A-2. We will also need the constant Ru
a=29.11
b=0.1916×102
c=0.4003×105
d=0.8704×109
Ru=8.31447kJkmolK
Step 2
To calculate the internal energy change per mole u we need to integrate the cv from the initial T1=200K to the final T2=800K temperature.
u=T1T2cv(T)dT
u=T1T2(aRu+bT+cT2+dT3)dT
u=T1T2(aRu+bT+cT2+dT3)dT
u=(aRu)T1T2dT+bT1T2TdT+cT1T2T2dT+dT1T2T3dT
nick1337

nick1337

Expert2021-12-27Added 777 answers

a) From Table A-2 C
cv=(aR)+bT+cT2+dT3
where: a=29.11
b=0.1916×102
c=0.4003×105
d=0.8704×109
Substituting:
u=(29.118.314)+(0.1916×102)(800200)+(0.4003×105)(80022002)+(0.8704×109)(80032003)
u=12487 kJ/kmol
u=6194 kJ/kg
b) From Table B-2
At 500 K, (average Temperature)
cv=10.839 kJ/kgK
u=cv(T2T1)
u=6233 kJ/kg
c) Table A-2a
cv=10.183 kJ/kgK
u=cv(T2T1)
u=6110 kJ/kg

Mr Solver

Mr Solver

Skilled2023-06-19Added 147 answers

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:
cp=a+bT+cT2+dT3
where cp 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:
Δu=T1T2cpdT
where T1 is the initial temperature (200 K) and T2 is the final temperature (800 K).
Integrating the specific heat capacity equation and evaluating it between the temperature limits, we have:
Δu=200800(a+bT+cT2+dT3)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 cv value at the average temperature (Table A-2b):
The cv value for hydrogen at the average temperature can be obtained from Table A-2b. Let's denote this value as cvavg.
The internal energy change can be calculated using the relationship:
Δu=cvavg·(T2T1)
where T1 is the initial temperature (200 K) and T2 is the final temperature (800 K).
Step 3:
(c) Using the cv value at room temperature (Table A-2a):
Similarly, the cv value for hydrogen at room temperature can be obtained from Table A-2a. Let's denote this value as cvroom.
The internal energy change can be calculated using the relationship:
Δu=cvroom·(T2T1)
where T1 is the initial temperature (200 K) and T2 is the final temperature (800 K).
madeleinejames20

madeleinejames20

Skilled2023-06-19Added 165 answers

To determine the internal energy change Δu 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:
cp=a+bT+cT2+dT3+eT4 where cp 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:
Δu=T1T2cpdT
Substituting the values of T1=200 K and T2=800 K into the equation, we have:
Δu=200800(a+bT+cT2+dT3+eT4)dT
(b) The cv value at the average temperature (Table A-2b):
The specific heat at constant volume, cv, is given as a constant value at different temperatures in Table A-2b. We can use the average value of cv over the temperature range to calculate the internal energy change.
(c) The cv value at room temperature (Table A-2a):
Similar to approach (b), we can use the cv 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 cv value at the average temperature:
From Table A-2b, we can find the cv 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:
Δu=cv·ΔT
(c) Using the cv value at room temperature:
From Table A-2a, we can find the cv 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:
Δu=cv·ΔT
Now you have the three different approaches to calculate the internal energy change for hydrogen as it is heated from 200 to 800 K.

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