Gregory Emery
2021-12-15
Answered

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).

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vrangett

Answered 2021-12-16
Author has **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

We have step-by-step solutions for your answer!

Marcus Herman

Answered 2021-12-17
Author has **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

We have step-by-step solutions for your answer!

nick1337

Answered 2021-12-27
Author has **575** answers

a) From Table A-2 C

where:

Substituting:

b) From Table B-2

At 500 K, (average Temperature)

c) Table A-2a

We have step-by-step solutions for your answer!

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