A sample of gas has its number of molecules halved, its Kelvin temperature doubled, and its volume halved. What is the new pressure, relative to the original pressure?

To determine the new pressure, we need to use the combined gas law equation:

P1V1/T1 = P2V2/T2

Where:
P1 = original pressure
V1 = original volume
T1 = original temperature (in Kelvin)
P2 = new pressure (to be determined)
V2 = new volume
T2 = new temperature (in Kelvin)

Given that the number of molecules is halved, the new volume is halved, and the Kelvin temperature is doubled, we can write the following relationships:

V2 = V1/2
T2 = 2T1

Now, substitute these values into the combined gas law equation:

P1V1/T1 = P2V2/T2

P1V1/T1 = P2(V1/2)/(2T1)

Simplifying further:

P1V1 = (P2V1)/(4T1)

Multiplying both sides by 4:

4P1V1 = P2V1

Canceling out V1:

4P1 = P2

Therefore, the new pressure (P2) is four times the original pressure (P1).

To determine the new pressure relative to the original pressure, we can use the ideal gas law. The ideal gas law equation is:

PV = nRT

Where:
P is the pressure
V is the volume
n is the number of moles
R is the ideal gas constant
T is the temperature in Kelvin

In this case, we want to compare the new pressure (P') to the original pressure (P).

Let's assume the original state is represented by variables P, V, n, and T, and the final state is represented by variables P', V', n', and T'.

According to the problem, the number of molecules is halved, so n' = n/2. The Kelvin temperature is doubled, so T' = 2T. The volume is halved, so V' = V/2.

Now, let's compare the equations for the original state and the final state:

Original state: PV = nRT

Final state: P'V' = (n/2)RT'

Substituting the known values for V', n', and T', we get:

P(V/2) = (n/2)R(2T)

Simplifying:

PV = nRT

Since the original state equation is the same as the final state equation, we can conclude that the new pressure (P') is equal to the original pressure (P).

Therefore, the new pressure is equal to the original pressure, and the new pressure is not relative to the original pressure but rather the same as the original pressure.

P V = n R T

well, R does not change :)
R = P V/(nT) = P' .5V /(.5n 2T) = P'(.5)V/(nT)
so
.5 P' = P
P' = 2 P