Explanation Write the given equation of state of air. P ( v − a ) = R T (I) Here, pressure is P , specific volume is v , universal gas constant is R , temperature is T and the constant is a . Rewrite Equation (I). P = R T v − a (II) Partially differentiate Equation (II) with respect to the temperature ( T ) by keeping the specific volume ( v ) as constant. ( ∂ P ∂ T ) v = R ( 1 ) v − a = R v − a Write the formula for change in internal energy ( d u ) given in Equation (12-29). d u = c ν d T + [ T ( ∂ P ∂ T ) v − P ] d v (III) Here, specific heat at constant volume is c ν , change in temperature is d T , change in specific volume is d v and symbol ∂ indicates the partial derivative of variables. Substitute R v − a for ( ∂ P ∂ T ) v and R T v − a for P in Equation (III). d u = c ν d T + [ T ( R v − a ) − R T v − a ] d v = c ν d T + ( R T v − a − R T v − a ) d v = c ν d T + ( 0 ) d v = c ν d T (IV) Integrate Equation (IV) with two states 1 and 2. u 2 − u 1 = c ν ( T 2 − T 1 ) (V) Here, change in internal energy is u 2 − u 1 and subscripts 1 and 2 indicates the two states of air. Write the formula for change in internal energy of ideal gas model. d u = c ν d T For two states, u 2 − u 1 = c ν ( T 2 − T 1 ) (VI) Refer Table A-2b, “Ideal-gas specific heats of various common gases”, obtain the specific heat of air at constant volume ( c ν ) corresponding to the average temperature of 300 ° C + 20 ° C 2 = 160 ° C ( 433 K ) - using interpolation method

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Chapter 12.6, Problem 34P

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The change in internal energy of air in kJ/kg and compare the result to the value obtained by using ideal gas equation of state.

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Chapter 12.6, Problem 33P

Chapter 12.6, Problem 35P

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