Ideal gas law - isovolumetric problem

[portofino]

Homework Statement

A gas sample enclosed in a rigid metal container at room temperature (20 C) has an absolute pressure p_1. The container is immersed in hot water until it warms to 40 C. What is the new absolute pressure p_2?

Express your answer in terms of p_1

Homework Equations

PV = nRT where P is pressure in pascals, V is volume, n is the number of moles, R is constant = 8.314, T is temperature in kelvin

convert celsius to kelvin
K=C+273.15 where K is temp in kelvin, and C is temp in celsius

The Attempt at a Solution

since this problem is isovolumetric, the volume remains constant.

p_1 = nRT_1/V where T_1 = 20C = 293 K assuming n,R,V are constant

p_2 = nRT_2/V where T_2 = 40C = 313 K assuming n, R, V are constant

how do i put p_2 in terms of p_1, do i just solve p_1 in terms of say V and substitute it in for V in the p_2 equation?

doing so i get:

V =nRT_1/p_1 = 293/p_1 asuming n and R are constant

thus substituting V for V in the p_2 equation i get:

p_2 = nRT_2/V = 313/(292/p_1) = 313p_1/292

is that correct? I'm almost certain it is not correct.

 

[LowlyPion]

Homework Statement

A gas sample enclosed in a rigid metal container at room temperature (20 C) has an absolute pressure p_1. The container is immersed in hot water until it warms to 40 C. What is the new absolute pressure p_2?

Express your answer in terms of p_1

Homework Equations

PV = nRT where P is pressure in pascals, V is volume, n is the number of moles, R is constant = 8.314, T is temperature in kelvin

convert celsius to kelvin
K=C+273.15 where K is temp in kelvin, and C is temp in celsius

The Attempt at a Solution

since this problem is isovolumetric, the volume remains constant.

p_1 = nRT_1/V where T_1 = 20C = 293 K assuming n,R,V are constant

p_2 = nRT_2/V where T_2 = 40C = 313 K assuming n, R, V are constant

how do i put p_2 in terms of p_1, do i just solve p_1 in terms of say V and substitute it in for V in the p_2 equation?

doing so i get:

V =nRT_1/p_1 = 293/p_1 asuming n and R are constant

thus substituting V for V in the p_2 equation i get:

p_2 = nRT_2/V = 313/(292/p_1) = 313p_1/292

is that correct? I'm almost certain it is not correct.

That looks right. For such a question though - where you have an isobaric or isothermal you can just use the simple ratio. (You can always divide an equation by an equation.)

$$\frac{P_1V_1}{P_2V_2} = \frac{nrT_1}{nrT_2}$$

In your case it Volume stays the same n the same and r the same so:

$$\frac{P_1}{P_2} = \frac{T_1}{T_2} = \frac{293}{313}$$

 

[Redbelly98]

Another way to set it up, and which may be easier to remember, is to solve the ideal gas equation for R:

$$R = \frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_2}$$

or in other words

$$\frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_2}$$

Then you cancel all the quantities that are equal (in this case V and n) and go from there.

 

[portofino]

the thing is when tried (313p_1)/292, it was incorrect and it stated that "your answer either contains an incorrect numerical multiplier or is missing one."

noticed how i entered the denominator as 292 as opposed to 293, would that be the reason why it is incorrect?

 

[Redbelly98]

It is incorrect by a very small amount, 0.3%. Why don't you try with the correct numbers and see what happens? Also, you could try dividing 313/293 on a calculator, and use that number times p_1.

 

[portofino]

yes i tried it with 293 as the denominator. it was correct

thanks for your help