Gas Pressure Formula

There are many different formulas that state the relationship between the pressure of the gas and temperature at different variants. But before learning about the gas pressure formula, it is important to know what is gas pressure. As all of these laws are related to the ideal gas and the equation derived from it. All the other laws are based on the relative temperature and pressure equation that is established with the ideal gas law. So, an ideal gas is a hypothetical situation where all the particles of the gas are colliding with each other and are in a perfect elastic state but do not possess any intermolecular attraction force between one another. 

So, the energy present in the ideal gas is in the form of kinetic energy only. And the internal energy that is present as kinetic energy only varies with the change in the temperature. Thus the characterization of the ideal gas is based on three variants and that is absolute pressure (P), absolute temperature (T) and volume of the gas (V). thus, according to ideal gas law, the gas pressure formula will be

PV = nRT = NkT or, P = nRT/V = NkT/ V

Where n is no. of moles of gas 

R is universal gas constant = 8.314 j/molK

N is no. of molecules present in gas

k is Bolttzmann’s constant = 1.38066 x 10-23 J/K or 8.617385 x 10-5 eV/K, where K=R/NA

NA is the avogadro’s number = 6.023 x 1023 / mol

Thus by this ideal gas formula, the gas law can be persuaded by rising the kinetic energy of the gas whose molecules collide with the walls of the container according to Newton's law. The average kinetic energy of the molecules is determined by their statistical elements. Hence the temperature becomes directly proportional to the average kinetic energy of the gas molecules. One mole of an ideal gas at STP will have a volume equals to 22.4 L. 

Solved Example :

Methane CH4 is used as an alternative fuel for automobiles replacing gasoline. If a gallon of gasoline is replaced by 665 grams of methane, then methane at 745 torr and 25oC will have how much of the volume?

From the ideal gas law the gas formula, PV = nRT or V = nRT / P

If R (universal gas constant) = 0.8206 L atm mol-1 K-1, thus all the amounts must be in moles, the temperature should be in kelvin and pressure must be in atm.

Therefore changing the units to the required units :

n = 665 g of CH4 x 1mol / 16.023g (molar mass of CH4)

   = 40.8 mol

1 atm pressure = 760 torr

Thus, 745 torr   = 745 torr x 1atm / 760 torr 

                            = 0.980 atm

Temperature from oC to K = 25 oC + 273 K = 298 K

Thus the required volume of methane is, V = nRT / P

= (40.8 mol x  0.8206 L atm mol-1 K-1 x 298K) / 0.980 atm

= 1.02 x 103 L = 1020 L or 269 gallon.

Amonton’s Law

Amonton’s law states that if the gas is kept at a constant volume and its number of moles remains constant that with the rise in the temperature the pressure of the gas also rises and vice versa. This can be illustrated with an example. If gas is filled in a container and is tightly secured with a cock so that the gas inside cannot escape or the gas outside cannot enter. Thus the volume and the no. of moles of gas will remain the same. Then the container is attached with a pressure gauge. Now, when the container is cooled, the gas inside is also cooled and its pressure drops. Now if the container is heated then the gas inside the container is also heated and therefore, its pressure also increases. The following illustrated diagram represents the same.

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Thus for a sample gas whose volume is kept constant, the temperature and pressure for the gas are directly proportional. Thus for a constant volume of any air, the temperature and the pressure will be in a linear relationship when the temperature is on the Kelvin scale. The measurement cannot be practically possible at a very low temperature as the air will undergo condensation at a very low temperature. Thus when the line in a graph represents the relationship between temperature and pressure of a gas at constant volume, the pressure reaches zero at -273oC of temperature that is 0 in kelvin scale which is the lowest possible temperature that could be reached. This is therefore called an absolute Zero. The below graph represents the same.

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Thus the mathematical representation of the linear relationship of temperature and pressure of a gas at constant volume according  to Amontan’s law is represented as 

P ∝ T or P = k xT, where, k is the proportionality constant that depends on the nature, amount and volume of the gas.  Thus for a confined gas at a constant volume will have temperature - pressure ratio that will be constant, that is, P/T = k (from the above gas formula of Amonton’s law). Therefore in the first condition of the confined gas the pressure and temperature ratio will be

P1 / T1 = k and in the second condition of the confined gas, the pressure and temperature ratio will be P2 / T2 = k. Thus from the first and second condition of the gas at a constant volume, the two ratios will be equal to each other. 

P1 / T1 = P2 / T2

This equation is important to find out the temperature and pressure of a confined gas at a constant volume in different conditions. 

Solved Example

A hairspray can that has isobutane gas as a propellant is used until it is completely empty. The temperature of the gas inside the hair spray was 24oC and the pressure was 360 kPa at the start of a hot summer day. If the can is exposed to a heated surface under direct sun heat and the temperature eventually rises to 50oC what will be the pressure given the volume of isobutane gas inside is 350 ml.

Like the P1 and the T1 of isobutane gas at 350 ml volume is given at an initial condition and the T2 is known. There using the gas formula of Amonton’s law the pressure for the second condition will be

 P1 / T1 = P2 / T2

360 kPa / 297 K (24oC) =  P2 / 323 K (50oC)

P2 = (P1 x T2) / T1 = (360 kPa x 323 K) / 297 K = 390 kPa

Boyle’s Law

According to Boyle’s Law, at a constant temperature when a confined gas occupies less volume it exerts more pressure and when the volume of the gas expands then the pressure exerted by the gas decreases proportionately. Thus unlike the pressure and temperature or volume and temperature relationship that is linear in nature the Volume and pressure of a confined gas are inversely proportional and has an exponential curve. This can be explained by an example where a syringe is partially filled with gas and is confined within the syringe and a pressure gauge is plugged on top of it. Now when the plunger is pushed downwards the volume of the gas decreases and it starts exerting more pressure and vice versa. This is illustrated by the following graphical representation of the same. 

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Thus, the mathematical expression of the gas pressure according to Boyle’s law will be 

P ∝ 1 / V or P = k x 1 / V or P x V = k

Therefore, from the above equation, it can be derived that P1 V1 = P2 V2

Where k is the universal gas constant.

This can be demonstrated by an illustration of a graph showing the inverse relationship between the pressure and the volume.

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Solved Example :

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The gas in the above graph has a volume of 15mL and the pressure is 13 psi. What will be the pressure of the gas at the volume 7.5mL using

1. P-V graph

2. 1 / P vrs. V graph

3. Boyle’s law

In the P - V graph, the pressure is around 27 psi. And in the 1 / P vrs. V graph, the pressure is around 26 psi.

Boyle’s law states that the product of pressure (P1 and P2) and volume (V1 and V2) of a confined gas at a constant temperature will be equal to the same value on gas constant, thus they are mutually equal to each other.  P1 V1 = k and P2 V2 = k, so P1 V1 = P2 V2

Thus P1 and V1 are 13 psi and 15 mL respectively. 

13 psi x 15 mL= P2 x 7.5 mL

P2 = (13 psi x 15 mL) / 7.5 mL = 26 psi

Charles’s Law

Charles’s law states that at a constant pressure of 1atm the volume of the gas increases with the increase in temperature and decreases with the decrease in the temperature. The V and T of a confined gas at constant pressure are mutually proportional to each other and the graph extrapolated will be linear in nature. 

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Mathematically, the gas formula according to the Charles law can be expressed as

 V ∝ T or V = k xT or V /T = k, thus for a constant pressure of a confined gas V1 / T1 = V2 / T2

Solved Example :

The sample carbon dioxide gas is at a constant pressure of 750 torr. At 10oC the volume of CO2 is 3.00 L. What will be the volume of CO2 at 30oC? 

According to the Charles law,  V1 / T1 = V2 / T2

Or,  3.00 L / 283 K =  V2 / 303 K

Or,  V2 = (3.00 L x 303 K) / 283 K = 0.321 L