How Is Volume Affected by Pressure in an Ideal Gas?

[MozAngeles]

Homework Statement

A monatomic ideal gas is held in a thermally insulated container with a volume of 0.0900m ³. The pressure of the gas is 110 kPa, and its temperature is 347 K.
To what volume must the gas be compressed to increase its pressure to 150 kPa?
To what volume must the gas be compressed to increase its pressure to 150 kPa?

Homework Equations

PV=nRT
$$\Delta$$=Q-W
W=P$$\Delta$$V

The Attempt at a Solution

P₁*V1/P₂
=(110*.09)/(150)
= .0660 this is wrong i don't know what I'm missing..

 

[Andrew Mason]

Homework Statement

A monatomic ideal gas is held in a thermally insulated container with a volume of 0.0900m ³. The pressure of the gas is 110 kPa, and its temperature is 347 K.
To what volume must the gas be compressed to increase its pressure to 150 kPa?
To what volume must the gas be compressed to increase its pressure to 150 kPa?

Homework Equations

PV=nRT
$$\Delta$$=Q-W
W=P$$\Delta$$V

The Attempt at a Solution

P₁*V1/P₂
=(110*.09)/(150)
= .0660 this is wrong i don't know what I'm missing..

The key is the thermally insulated container. What kind of compression is this? What is the relationship between P and V in such a compression? (hint: it has something to do with $\gamma$)

AM

 

[MozAngeles]

So it is a adiabatic compression right?
So would I use
P_iV_i^$$\gamma$$=P_fV_f^$$\gamma$$

 

[Andrew Mason]

So it is a adiabatic compression right?
So would I use
P_iV_i^$$\gamma$$=P_fV_f^$$\gamma$$

If you mean:

$$P_iV_i^{\gamma} = P_fV_f^{\gamma}$$

ie: $$PV^{\gamma} = K = constant$$

then, yes

AM

 

[MozAngeles]

and then for the second part of the question I am still stumped, I thought you could use
V_i/T_i=V_f/T_f

and this isn't right, probably because of the fact that it is adiabatic?
but the equation is PV^$$\gamma$$= constant
so that doesn't include temperature, and now I'm lost..

 

[Andrew Mason]

and then for the second part of the question I am still stumped, I thought you could use
V_i/T_i=V_f/T_f

and this isn't right, probably because of the fact that it is adiabatic?
but the equation is PV^$$\gamma$$= constant
so that doesn't include temperature, and now I'm lost..

V_i/T_i=V_f/T_f is true only if P is constant. In an adiabatic change, P, V and T all change. The ideal gas law still applies. But in order to determine how T changes you have to know how P and V change.

If you substitute P = nRT/V into the adiabatic condition, it becomes:

$$TV^{(\gamma-1)} = PV^\gamma/nR = K/nR = \text{constant}$$

That is what you have to use.

AM

 

[MozAngeles]

thanks figured it out