Considering the ideal gas law PV = nRT, what is P directly proportional to?

Answer 1

Here's what's going on here.

First, confirm that you truly understand what is meant by directly proportional.

Two quantities must either increase as the other increases or decrease as the other decreases, both at the same rate, for them to be considered directly proportional.

Starting from the ideal gas law equation, isolate #P# on one side of the equation first
#PV = nRT implies P = (nRT)/V#
Since #R# represents a constant, you can write it separately
#P = (nT)/V * R#
Now focus on the ratio that exists between number of moles, #n#, and temperature, #T#, on one side, and volume, #V#, on the other.

Let's start with the number of moles. You must maintain the other two variables constant in order to determine whether there is direct or inverse proportionality.

#P = n * overbrace(T/V * R)^(color(blue)("constant"))#
So, under these circumstance, what would happen to #P# if #n# were to increase? Well, in order for the equal sign to remain valid, #P# would have to Increase as well.
Likewise, if #n# were to decrease, #P# would have to decrease as well. Therefore, you can say that

As long as the temperature and volume remain constant, the number of moles and the pressure are exactly equal.

The same can be said for temperature, #T#. Keeping the other two variables constant will result in
#P = T * overbrace(n/V * R)^(color(blue)("constant"))#

It can be stated that once more, a rise in temperature will lead to an increase in pressure, and a fall in temperature will lead to a decrease in pressure.

According to Gay Lussac's Law, pressure and temperature are directly correlated when the number of moles and volume are held constant.

Lastly, maintain a constant temperature and mole count while observing the effects of volume variation on pressure.

#P = 1/V * overbrace(n * T * R)^(color(blue)("constant"))#

Since volume is now in the denominator, a rise in volume would cause a drop in pressure this time.

#V uarr implies 1/V darr#

Thus, when the number of moles and temperature are held constant, pressure is not directly proportional to volume. Conversely, a decrease in volume would lead to an increase in pressure.

Nonetheless, you could say that

Boyle's Law states that when temperature and the number of moles are held constant, pressure is inversely proportional to volume.

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Answer 2

Pressure (P) is directly proportional to temperature (T) when volume (V), amount of substance (n), and the gas constant (R) are held constant.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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