### Video Transcript

Which of the following is the
correct formula relating the pressure and temperature of a gas that is kept at a
constant volume? 𝑇 represents the temperature of
the gas, 𝑝 represents the pressure of the gas, and 𝑉 represents the volume of the
gas. (A) 𝑝 times 𝑇 equals a
constant. (B) 𝑝 times 𝑇 squared equals a
constant. (C) 𝑝 times 𝑉 equals 𝑇. (D) 𝑝 over 𝑇 equals a
constant. (E) 𝑝 over 𝑇 equals 𝑉.

Alright, so in this question, we’re
given five different potential formulas that we’re told could relate the pressure
and temperature of a gas that’s kept at a constant volume. Our task is to work out which of
these formulas is the correct one. Now, one necessary condition for a
formula to be correct is that the units on the left- and the right-hand side of it
must agree with each other. This isn’t going to help us with
the formulas presented in options (A), (B), and (D) because these all contain a
constant on the right-hand side, which could have any kind of units. However, it can help us work out
whether option (C) or (E) could be correct.

The three quantities in these
formulas are 𝑝, the pressure; 𝑉, the volume; and 𝑇, the temperature of the
gas. So let’s recall what the units are
for each of these quantities. The SI unit for pressure is the
pascal, which is equal to the newton per meter squared. Since Newton’s second law states
that force is equal to mass multiplied by acceleration, and we can recall that force
has SI units of newtons, mass has SI units of kilograms, and acceleration has SI
units of meters per second squared, then this means that units of newtons must be
equivalent to units of kilograms meters per second squared. Overall then, we have that pascals,
the unit for pressure, can be expressed in SI base units as kilograms meters to the
minus one seconds to the minus two. Meanwhile, the units for volume are
cubic meters, so that’s meters to the power of three. Lastly, temperature is measured in
units of kelvin.

So we’ve now got the units for each
of these three quantities expressed in terms of SI base units. If we consider the units in option
(C), then on the left-hand side, that’s units of pressure multiplied by volume. And that’s the same thing as units
of pressure multiplied by units of volume. Substituting in our expressions for
the units of pressure and volume and then multiplying out the meters to the minus
one and the meters cubed, we find that the left-hand side of the equation has units
of kilograms meters squared per second squared.

Meanwhile, the units on the
right-hand side of the formula are just units of temperature. And we know that the SI base unit
for temperature is units of kelvin. Clearly, the units of kelvin on the
right-hand side don’t agree with the units of kilograms meters squared per second
squared on the left-hand side. That means that the formula given
in option (C) cannot be correct.

Now, let’s consider option (E),
which says 𝑝 divided by 𝑇 is equal to 𝑉. Now, we could do the same process
that we just did for the formula in option (C). However, we can notice that the
right-hand side of the formula, which is just equal to the volume 𝑉, must have SI
base units of meters cubed. Meanwhile, on the left-hand side,
there’s a temperature term in the denominator, which has units of kelvin. And since kelvin don’t enter into
the expression for the units of pressure, then there’s no way that this kelvin in
the denominator can cancel out. So we know that within the units on
the left-hand side, there’s a per kelvin, but there’s no kelvin at all in the units
on the right. That means that there’s no way for
the units on the left and the right to agree. So the formula in option (E) cannot
be correct.

At this stage, we’ve eliminated two
of the potential formulas. And this is as far as we can get by
considering the units. To decide between the three
remaining available options, we need to consider the physics of the situation. We’re told that we’ve got a gas
that’s being kept at a constant volume. So we could imagine, for example,
that it’s being kept inside a box with fixed walls. Since the substance is a gas, then
it’s made up of a load of particles that are free to move around. And that’s exactly what’s going
on. All of the particles are flying
around in all different directions inside the box. And as they do this, they can
collide with each other and importantly also with the walls of the box.

Supposing that these blue arrows
we’ve drawn represent the particles’ velocities at some instant in time, then we can
see that this particle in the top right is about to hit this right-hand side of the
box. Now, since the volume of this box
is fixed, then we know that the walls can’t move. So when this particle collides with
the wall, it will then bounce off the wall and the direction of its velocity will
have changed. During this collision, the particle
will exert a force on the wall, and there’ll be a component of this force that acts
outward.

In this sketch, we’ve just drawn a
small number of particles to give an idea of what’s going on. But in reality, we would expect
there to be a far larger number of particles inside the box. That means that there’s going to be
lots of collisions between these particles and the walls of the box. And so there’ll be lots of these
forces, each with an outward component, acting on the walls. These outward forces acting over
the area of the box walls result in a pressure on the walls of the box.

Now, the faster the particles in
the gas are moving, the more force they’re going to exert on the walls when they
collide with them. And so the faster the particles,
the greater the pressure exerted on the walls. We can recall that the particles in
a gas move with some average speed and that this average speed indicates the
temperature of the gas. Specifically, the greater this
average speed or the faster the particles, the greater the temperature of the
gas. Since a greater average speed of
the particles means that the gas has a higher temperature and we found that
faster-moving particles will result in a greater pressure, then we can combine these
two statements to say that higher-temperature particles will exert a greater
pressure.

If we look at the formulas given in
options (A) and (B), we can see that we’ve got pressure times temperature equals a
constant or pressure times temperature squared equals a constant. If we have 𝑝 multiplied by 𝑇
equals a constant, as in option (A), then if the temperature 𝑇 increases, the
pressure 𝑝 must decrease in order for their product to remain constant. But what we found by considering
the physics of the situation is that if the temperature 𝑇 increases, then the
pressure 𝑝 must also increase. So then the formula in option (A)
cannot be correct.

The same thing is true for option
(B), which says that pressure times temperature squared equals a constant. Because again if the temperature 𝑇
increases in this formula, then the pressure 𝑝 would have to decrease in order for
the product 𝑝 times 𝑇 squared to remain constant. So then we can also eliminate
option (B).

This just leaves us with the
formula from option (D), which says that 𝑝 divided by 𝑇 is equal to a
constant. Now, we have reasoned that if we’ve
got a gas kept at a constant volume and we increase its temperature 𝑇, then the
pressure 𝑝 must also increase. The exact relationship between
pressure and temperature was discovered back in the 19th century and is known as
Gay-Lussac’s law. This law says that pressure is
directly proportional to temperature, which means that if the temperature of a gas
increases, then the pressure will increase in proportion to that. This can equivalently be written as
pressure is equal to a constant multiplied by temperature.

We can see that if we take this
equation and divide both sides of it by the temperature 𝑇, then on the right-hand
side, the 𝑇 in the numerator cancels with the one in the denominator. That means that Gay-Lussac’s law
can be written as 𝑝 divided by 𝑇 is equal to a constant, which is exactly the
formula we’ve got in option (D). So then this formula from option
(D) is our answer to the question. The correct formula relating the
pressure and temperature of a gas that’s kept at a constant volume is 𝑝 divided by
𝑇 is equal to a constant.