A closed thermodynamic system initially has internal energy and volume

PHYSICS · Multi-part · A Level · Physics · Engineering physics · Engineering physics

Problem

A closed thermodynamic system initially has internal energy

U_0

and volume

V_0

. From

t = 0

t = T

the following occur simultaneously: • An electrical heater supplies energy to the system at a constant rate (power)

P

. • The system pushes against a constant external pressure

P_{\text{ext}}

, and the volume increases uniformly with time so that at

t = T

the volume is

V_0 + \Delta V

. Assume no other energy transfers occur. (a) Derive an algebraic expression for the mechanical work done on the system during the time interval

0

T

in terms of

P_{\text{ext}}

and

\Delta V

. (b) In terms of

U_0

P

T

P_{\text{ext}}

, and

\Delta V

, obtain an expression for the internal energy

U_T

of the system at time

T

. (c) Evaluate

U_T

for the case

U_0 = 1.2\times10^{3}\,\text{J}

P = 50\,\text{W}

T = 30\,\text{s}

P_{\text{ext}} = 1.5\times10^{5}\,\text{Pa}

, and

\Delta V = 1.2\times10^{-3}\,\text{m}^3

. Show all steps and report the final answer with appropriate units.