Final Temperature and Pressure Increase in Adiabatic Air Compression?

[rg2004]

I'd just like to check my answer, it seems too big

Homework Statement

Air initially at 20◦C is compressed by a factor of 15.

What is the final temperature assuming that the compression is adiabatic and $$\gamma$$ ≈ 1.4 the value of $$\gamma$$ for air in the relevant range of temperatures? By what factor does the pressure increase?

Homework Equations

PV^$$\gamma$$=constant
TV^$$\gamma$$-1=constant

The Attempt at a Solution

V₁/15=V₂
T₁V₁^$$\gamma$$-1=T₂V₂^$$\gamma$$-1
293.15*V₁^$$\gamma$$-1=T₂*V₁/15
293.15*15^$$\gamma$$-1=T₂
293.15*15^.4=T₂
T₂=866.017 kelvinP₁V₁^$$\gamma$$=P₂V₂^$$\gamma$$
P₁V₁^$$\gamma$$=P₂V₁/15^$$\gamma$$
P1*15^$$\gamma$$=P2
thus P2 increased by a factor of 44.31does it look right?

 

[Andrew Mason]

Looks right. You can also use the ideal gas law to find final pressure:

$$nR = P_1V_1/T_1 = P_2V_2/T_2$$

$$P_2/P_1 = T_2V_1/T_1V_2 = \frac{866*15}{293} = 44.3$$

AM

 

[rg2004]

Thanks!