4.6 Key equations

4. The First Law of Thermodynamics for Closed Systems

Constant-volume specific heat

C_{v} = {(\frac{\partial u}{\partial T})}_{v}

Change in specific internal energy for all fluids

Δ u = u_{2} - u_{1}

Change in specific internal energy for ideal gases

Δ u = C_{v} (T_{2} - T_{1})

Specific heat transfer

q = \frac{Q}{m}

Boundary work

_{1} W_{2} = \int_{1}^{2} P d V

Specific boundary work

_{1} w_{2} = \int_{1}^{2} P d v

Spring force

F = K x

Spring work

W_{s p r i n g} = \int_{1}^{2} F d x = \frac{1}{2} K (x_{2}^{2} - x_{1}^{2})

The first law of thermodynamics for closed systems

Δ U = U_{2} - U_{1} =_{1} Q_{2} -_{1} W_{2}

, assuming

Δ K E = Δ P E = 0

Equations for polytropic Processes

Process function

P v^{n} = c o n s t a n t

Boundary work for real gases

If

n \neq 1

,

$_{1} W_{2} = \frac{P_{2} V_{2} - P_{1} V_{1}}{1 - n}$

If $n = 1,$

$_{1} W_{2} = P_{1} V_{1} l n \frac{V_{2}}{V_{1}} = P_{2} V_{2} l n \frac{V_{2}}{V_{1}}$

$_{1} W_{2} = P_{1} V_{1} l n \frac{P_{1}}{P_{2}} = P_{2} V_{2} l n \frac{P_{1}}{P_{2}}$

Specific boundary work for real gases

If

n \neq 1

$_{1} w_{2} = \frac{P_{2} v_{2} - P_{1} v_{1}}{1 - n}$

If $n = 1,$

$_{1} w_{2} = P_{1} v_{1} l n \frac{v_{2}}{v_{1}} = P_{2} v_{2} l n \frac{v_{2}}{v_{1}}$

$_{1} w_{2} = P_{1} v_{1} l n \frac{P_{1}}{P_{2}} = P_{2} v_{2} l n \frac{P_{1}}{P_{2}}$

Boundary work for ideal gases

If

n \neq 1

$_{1} W_{2} = \frac{P_{2} V_{2} - P_{1} V_{1}}{1 - n}$

If $n = 1,$

$_{1} W_{2} = P_{1} V_{1} l n \frac{V_{2}}{V_{1}} = P_{2} V_{2} l n \frac{V_{2}}{V_{1}}$

$_{1} W_{2} = P_{1} V_{1} l n \frac{P_{1}}{P_{2}} = P_{2} V_{2} l n \frac{P_{1}}{P_{2}}$

$_{1} W_{2} = m R T l n \frac{V_{2}}{V_{1}} = m R T l n \frac{P_{1}}{P_{2}}$

( $T$ in Kelvin)

Specific boundary work for ideal gases

If

n \neq 1

$_{1} w_{2} = \frac{P_{2} v_{2} - P_{1} v_{1}}{1 - n}$

If $n = 1,$

$_{1} w_{2} = P_{1} v_{1} l n \frac{v_{2}}{v_{1}} = P_{2} v_{2} l n \frac{v_{2}}{v_{1}}$

$_{1} w_{2} = P_{1} v_{1} l n \frac{P_{1}}{P_{2}} = P_{2} v_{2} l n \frac{P_{1}}{P_{2}}$

$_{1} w_{2} = R T l n \frac{v_{2}}{v_{1}} = R T l n \frac{P_{1}}{P_{2}}$

( $T$ in Kelvin)

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