Calculating final pressure and temperature of cylinder

[mh1985]

Homework Statement

Using a value of n of 1.1, calculate the final pressure and temperature when the cylinder is compressed rapidly with a compression ratio of 15.

Starting pressure = 101.325 kPa
Starting temp = 298 K

Homework Equations

(V2/V1) = 15

The Attempt at a Solution

Not sure about this part

 

[rock.freak667]

Rapid compression would mean that the compression is adiabatic. So you also know what other equation?

The compression ratio would be P₂/P₁=15 and you know what P₁ is.

Using the above equation you found and the ideal gas equation, can you form an equation involving only P and T?

 

[mh1985]

Rapid compression would mean that the compression is adiabatic. So you also know what other equation?

The compression ratio would be P₂/P₁=15 and you know what P₁ is.

Using the above equation you found and the ideal gas equation, can you form an equation involving only P and T?

Thanks for the reply,

So I multiply both sides of P₂/P₁=15 by P₁ to get P₂,

101325 * 15 = P2 = 1519875 Pa

Not sure how to form the equation involving only P & T...

Something like T₂/T₁ = (P₂/P₁)^((n-1)/n)

EDIT:

so (P₂/P₁)^((n-1)/n) = 15^(1.1 - 1)/1.1 = 1.2791

1.2791 = T₂/T₁

T₁*1.2791 = T₂ = 381.171 K

Not sure if this is right?

 

[rock.freak667]

Thanks for the reply,

So I multiply both sides of P₂/P₁=15 by P₁ to get P₂,

101325 * 15 = P2 = 1519875 Pa

Not sure how to form the equation involving only P & T...

Something like T₂/T₁ = (P₂/P₁)^((n-1)/n)

EDIT:

so (P₂/P₁)^((n-1)/n) = 15^(1.1 - 1)/1.1 = 1.2791

1.2791 = T₂/T₁

T₁*1.2791 = T₂ = 381.171 K

Not sure if this is right?

Well you'd have P₁Vⁿ₁=P₂V₂² and P₁V₁/T₁ = P₂V₂/T₂


Take the last equation, make V₁ the subject and substitute it into the first,the V₂ should at least cancel out if I remember correctly.

 

[mh1985]

Well you'd have P₁Vⁿ₁=P₂V₂² and P₁V₁/T₁ = P₂V₂/T₂


Take the last equation, make V₁ the subject and substitute it into the first,the V₂ should at least cancel out if I remember correctly.

Hi thanks for the reply

If I make V₁ the subject, I get P₂V₂T₁ /P₁T₂, right so far? :S

But I don't see how to substitute it into the first equation?

P₁ * (P₂V₂T₁ /P₁T₂) = P₂V₂^2

 

[rock.freak667]

Hi thanks for the reply

If I make V₁ the subject, I get P₂V₂T₁ /P₁T₂, right so far? :S

But I don't see how to substitute it into the first equation?

P₁ * (P₂V₂T₁ /P₁T₂) = P₂V₂^2

Sorry, I meant your equation to sub into should be

P₁V₁ⁿ=P₂V₂ⁿ

Just sub your first equation and the volume should cancel out.

 

[mh1985]

Sorry, I meant your equation to sub into should be

P₁V₁ⁿ=P₂V₂ⁿ

Just sub your first equation and the volume should cancel out.

So I have:

P₁* (P₂T₁/P₁T₂)ⁿ= V₂ ?

 

[rock.freak667]

So I have:

P₁* (P₂T₁/P₁T₂)ⁿ= V₂ ?

You have

$$V_1= \frac{P_2 V_2 T_1}{T_2 P_1}$$

and

$$P_1V_1^n = P_2V_2^n$$

so that

$$P_1 \left( \frac{P_2 V_2 T_1}{T_2 P_1} \right)^n = P_2V_2^n$$

You can see that V₂ⁿ is common to both sides and it will cancel out leaving you with only pressure and temperature variables. Thus you can easily get T₂ with some simple algebra.