Gregory Emery

2021-12-15

Determine the internal energy change $\mathrm{\u25b3}u$ 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 $c}_{v$ value at the average temperature (Table A-2b), and (c) the $c}_{v$ value at room temperature (Table A-2a).

vrangett

Beginner2021-12-16Added 36 answers

Step 1

Given:

- Initial temperature${T}_{1}=200K$

- Final temperature${T}_{2}=800K$

Required

- Determine the internal energy change of hydrogen using

a) The empirical specific heat equation

b) The$c}_{v$ value at average temperature

c) The$c}_{v$ value at room temperature

Step 2

Solution

Part a

- Using the empirical relation of${\stackrel{\u2015}{c}}_{p}\left(T\right)$ from table (A-2C) and relating it to ${\stackrel{\u2015}{c}}_{v}\left(T\right)$

$\stackrel{\u2015}{c}}_{v}\left(T\right)={\stackrel{\u2015}{c}}_{p}-{R}_{u}=(a-{R}_{u})+bT+c{T}^{2}+d{T}^{3$

Where$(a=29.11,b=-0.1916\times {10}^{-2},c=0.4003\times {10}^{-5},d=-0.8704\times {10}^{-9})$

- The internal energy change could be defined as the following

$\mathrm{\u25b3}\stackrel{\u2015}{u}={\int}_{1}^{2}{\stackrel{\u2015}{c}}_{v}\left(T\right)dT={\int}_{1}^{2}[(a-{R}_{u})+bT+c{T}^{2}+d{T}^{3}]dT$

$\mathrm{\u25b3}\stackrel{\u2015}{u}=(a-{R}_{u})({T}_{2}-{T}_{1})+\frac{1}{2}b({T}_{2}^{2}-{T}_{1}^{2})+\frac{1}{3}c({T}_{2}^{3}-{T}_{1}^{3})+\frac{1}{4}d({T}_{2}^{4}-{T}_{1}^{4})$

$\mathrm{\u25b3}\stackrel{\u2015}{u}=(29.11-8.314)(800-200)+\frac{1}{2}\times (-0.1916\times {10}^{-2})\times ({800}^{2}-{200}^{2})+\frac{1}{3}\times (0.4003\times {10}^{-5})\times ({800}^{3}-{200}^{3})+\frac{1}{4}\times (-0.8704\times {10}^{-9})\times ({800}^{4}-{200}^{4})=12487K\frac{J}{K}mol$

Given:

- Initial temperature

- Final temperature

Required

- Determine the internal energy change of hydrogen using

a) The empirical specific heat equation

b) The

c) The

Step 2

Solution

Part a

- Using the empirical relation of

Where

- The internal energy change could be defined as the following

Marcus Herman

Beginner2021-12-17Added 41 answers

Step 1

a) In this problem we need to determine the internal energy change$\mathrm{\u25b3}u$ by using three different methods.

The first method is with the empirical specific heat equation. The equation gives us$c}_{p$ and we need the $\stackrel{\u2015}{{c}_{v}}$ for the calculation.

$\stackrel{\u2015}{{c}_{p}}=a+bT+c{T}^{2}+d{T}^{3}$

$\stackrel{\u2015}{{c}_{v}}=\stackrel{\u2015}{{c}_{p}}-{R}_{u}$

$\stackrel{\u2015}{{c}_{v}}=a-{R}_{u}+bT+c{T}^{2}+d{T}^{3}$

The values for the constants a,b,c and d we find in the table A-2. We will also need the constant$R}_{u$

$a=29.11$

$b=-0.1916\times {10}^{-2}$

$c=0.4003\times {10}^{-5}$

$d=-0.8704\times {10}^{-9}$

$R}_{u}=8.31447\frac{kJ}{kmolK$

Step 2

To calculate the internal energy change per mole$\mathrm{\u25b3}\stackrel{\u2015}{u}$ we need to integrate the $c}_{v$ from the initial ${T}_{1}=200K$ to the final ${T}_{2}=800K$ temperature.

$\mathrm{\u25b3}\stackrel{\u2015}{u}={\int}_{{T}_{1}}^{{T}_{2}}\stackrel{\u2015}{{c}_{v}}\left(T\right)dT$

$\mathrm{\u25b3}\stackrel{\u2015}{u}={\int}_{{T}_{1}}^{{T}_{2}}(a-{R}_{u}+bT+c{T}^{2}+d{T}^{3})dT$

$\mathrm{\u25b3}\stackrel{\u2015}{u}={\int}_{{T}_{1}}^{{T}_{2}}(a-{R}_{u}+bT+c{T}^{2}+d{T}^{3})dT$

$\mathrm{\u25b3}\stackrel{\u2015}{u}=(a-{R}_{u})\cdot {\int}_{{T}_{1}}^{{T}_{2}}dT+b\cdot {\int}_{{T}_{1}}^{{T}_{2}}TdT+c\cdot {\int}_{{T}_{1}}^{{T}_{2}}{T}^{2}dT+d\cdot {\int}_{{T}_{1}}^{{T}_{2}}{T}^{3}dT$

a) In this problem we need to determine the internal energy change

The first method is with the empirical specific heat equation. The equation gives us

The values for the constants a,b,c and d we find in the table A-2. We will also need the constant

Step 2

To calculate the internal energy change per mole

nick1337

Expert2021-12-27Added 777 answers

a) From Table A-2 C

where:

Substituting:

b) From Table B-2

At 500 K, (average Temperature)

c) Table A-2a

Mr Solver

Skilled2023-06-19Added 147 answers

madeleinejames20

Skilled2023-06-19Added 165 answers

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