Problem solving techniques: comparing two cases.
There are a couple of equivalent approaches to solving questions such as
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At what temperature will a helium atom have twice the average velocity that it has
at 300K ?
A sample of gas expands at constant pressure from 1 liter to 2 liters. What is the
final temperature if the beginning temperature is 200K?
A satellite orbiting a planet experiences a gravitational force of 400N. If the
satellite is moved to twice the distance from the planet, what is the new force of
gravity it experiences?
and so on......
Let me start with an ideal gas law example.
A sealed sample of gas in a rigid container at one atmosphere is
heated from 200K to 300K. What is the new pressure at 300K?
The ideal gas law applies: PV = nRT , but we don’t know either the volume or
the number of moles of gas. So direct calculation of the pressure this way
is not possible.
Approach 1 : isolate the varying quantities, set the resulting expression in
one case equal to the expression for the other case.
Sealed sample => n is constant.
Rigid container => constant volume
Gas constant, R, is always constant.
P = n R = constant
T
V
PV =nRT
The constant on the right is the same for both cases, so
P1 = n R = P2
or
P1 = P2
T1
V T2
T1 T2
Since we know P1 = 1 atm. T1 = 200K and T2 = 300K, we can calculate P2
1 atm. = P2
200K
300K
P2 = (1 atm) 300K/200K = 1.5 atm.
1
Notice that as long as the units are the same for the variables in the two
cases, they do not need to be changed to any standard units, they will cancel
as above. The one change you must make is to be sure temperature is in
Kelvin (or some other absolute scale, such as Rankine).
Approach 2 : divide equations.
P1 V1 = n1 R T1
P2 V2 = n2 R T2
In this problem, n, V and R are constant, so they divide out top and bottom
to give
P1 = T1
P2 T2
1 atm = 200K to give P2 = 1.5 atm.
P2
300K
For the molecular speed problem, this would look like:
v1 = √(3kT1/m)
v2 = √(3kT2/m)
this gives v1 = √T1
v2 √T2
2