When we combine the three gas laws into one expression:

V is directly proprtional to T
P is directly propetional to T ---------> PV is directly proprtional to T
P is inversely proportional to 1/V

PV/T= constant

*******I do not understand how PV is directly proportional to T and how PV/T= constant. Is there a mathematical manipulation or something? Please explain.

PV=nRT replacing nR with c,a constant, then PV=cT

so PV is directly proportional to T

for part 1, divide
PV=cT by T
PV/T=constant

To understand why PV is directly proportional to T and why PV/T is a constant, let's break it down step by step.

First, let's consider the three gas laws:

1. Boyle's Law: P is inversely proportional to V (P ∝ 1/V), where P is the pressure and V is the volume.

2. Charles's Law: V is directly proportional to T, where V is the volume and T is the absolute temperature (in Kelvin).

3. Gay-Lussac's Law (or Amonton's Law): P is directly proportional to T, where P is the pressure and T is the absolute temperature (in Kelvin).

Now, let's combine these three laws into one expression:

Since P is inversely proportional to V (P ∝ 1/V) and V is directly proportional to T (V ∝ T), we can write:

P ∝ 1/V --> P ∝ T/V (multiplied both sides by T)

Next, since P is directly proportional to T (P ∝ T), we can write:

P = kT/V (where k is a constant)

Now, we have an expression that relates P, V, and T. To eliminate the fraction, we can multiply both sides of the equation by V:

PV = kT

Here, PV is directly proportional to T. This means that if we increase the temperature (T), the product of pressure (P) and volume (V) will increase, and vice versa.

Finally, let's rearrange the equation further:

PV/T = k

Here, PV/T is a constant, which means that for a given amount of gas, the ratio of pressure times volume to temperature remains constant. This is known as the combined gas law.

So, mathematical manipulation and the combination of the three gas laws led us to the expression PV/T = constant, which indicates the relationship between pressure, volume, and temperature of a gas.