Classwork Series and Exercises {Chemistry – SS1}: Gas Law

Chemistry SS1 Week 8

Topic: Gas Law

Introduction

Gas has existed since the beginning of time; oftentimes, it was referred to as “air” or “oxygen;” however, in the late 18th century, “air” became known as gas, and people were able to distinguish between different types of gas. Towards the end of the 18th century, scientists started testing and developing laws that later became known as the “gas laws.” One of the most amazing things about gases is that, despite wide differences in chemical properties, all the gases more or less obey the gas laws.  These laws describe properties of gases, that is how gases behave with respect to pressure, volume, temperature and how they react in different situations. In order to understand the gas laws, we need to define a few terms:

Gas: A substance consisting of widely spread particles; it can expand indefinitely. This is also the third state of matter; it is not a solid or a liquid.

Pressure: The measure of force applied by another substance (such as a gas). It is commonly abbreviated as “P” (a capital letter P). Pressure can be measured in millimeters of Mercury (mmHg), torr, atmospheres (atm), Pascals (Pa), and kilopascals (kPa). All of the following measurements are the same, just different units, so you can use them to convert from one to the other. For the ideal gas law, the pressure will need to be in atmospheres. The conversions between these are as follows:

760 mmHg = 760 torr = 1.00 atm = 101,325 Pa = 101.325 kPa

Volume: The numerical amount of space occupied by a solid, liquid, or gas. It is commonly abbreviated as “V” (a capital letter V). Volume, in this situation, will be most often measured in liters, L.

Temperature: The measurement of the amount of energy seen in the motion of particles in a solid, liquid or gas. It can be measured on three scales: Fahrenheit, Celsius (sometimes referred to as Centigrade) and Kelvin. It is commonly abbreviated as “T” (a capital letter T). Temperature, in this situation, will most often be measured in Kelvin, K.

STP: STP stands for “standard temperature and pressure” and refers to conditions of 273 K (0 degrees C) and 1 atm.

The Gas Laws: Pressure Volume Temperature Relationships

Boyle’s Law:  The Pressure-Volume Law

Boyle’s law or the pressure-volume law states that the volume of a given amount of gas held at constant temperature varies inversely with the applied pressure when the temperature and mass are constant.

PV = C (C = Constant)

When pressure goes up, volume goes down. When volume goes up, pressure goes down.
From the equation above, this can be derived:

P₁V₁ = P₂V₂

Or like this:

Example 1: If the initial volume was 500 mL at a pressure of 760 atm, when the volume is compressed to 450 ml, what is the pressure?
Solution: V₁ = 500mL, P₁ = 760atm, V₂ = 450mL, P₂ = ?

P₁V₁ = P₂V₂

(760)(500) = P₂(450)
760 x 500 / 450 = P₂

P₂ = 844 atm
The pressure is 844 atm after compression.

Example 2: A 17.50cm³ sample of gas is at 4.50 atm. What will be the volume if the pressure becomes 1.50 atm, with a fixed amount of gas and temperature?

Solution: V₁ = 17.50cm3, P₁ = 4.50atm, V₂ = ?, P₂ = 1.50atm

P₁V₁ = P₂V₂

(4.50)(17.50) = (1.50)V₂

4.50 x 17.50 / 1.50 = V₂

V₂ = 52.50cm³

Example 3: A sample of air occupies a volume of 450 L at 20 ºC and 100 mmHg. What will be the pressure of this gas if it is transferred to a 200 L bulb at the same temperature?

Solution: V₁ = 450 L, P₁ = 100 mmHg, P₂ = ?, V₂ = 200 L

(100)(450) = P₂(200)

100 x 450 / 200 = P₂

P₂ = 225 L

 

This law states that the volume of a given amount of gas held at constant pressure is directly proportional to the temperature (Kelvin).

V T

Same as before, a constant can be put in:

V / T = C (C – Constant)

As the volume goes up, the temperature also goes up, and vice-versa.
Also same as before, initial and final volumes and temperatures under constant pressure can be calculated.

V₁ / T₁ = V₂ / T₂

Or like this:

Example 1: A sample of Carbon dioxide in a pump has volume of 20.5 cm³ and it is at 40 ^oC. When the amount of gas and pressure remain constant, find the new volume of Carbon dioxide in the pump if temperature is increased to 65 ^oC.

Solution: V₁ = 20.5 cm³, T₁ = 40 ^oC = (273 + 40)K, V₂ = ?, T₂ = 65 ^oC = (273 + 65)K

V₁ / T₁ = V₂ / T₂

V₂ = V₁.T₂ / T₁

V₂ = 20.5 x 338 / 313

= 22.1 cm³

Example 2: A sample of gas occupies 400.0 mL at 25.00 ^oC and 1 bar pressure. What volume will it occupy at 200.00 ^oC at the same P?

Solution: V₁ = 400.0 mL, T₁ = 25.00 ^oC = (25.00 + 273), V₂ = ?, T₂ = 200.00 ^oC = (200.00 + 273)K

Gay-Lussac’s Law:  The Pressure Temperature Law

This law states that the pressure of a given amount of gas held at constant volume is directly proportional to the Kelvin temperature.

P T

Same as before, a constant can be put in:

P / T = C

As the pressure goes up, the temperature also goes up, and vice-versa.
Also same as before, initial and final volumes and temperatures under constant pressure can be calculated.

P₁ / T₁ = P₂ / T₂

Example 1: Find the final pressure of gas at 150 K, if the pressure of gas is 210 mmHg at 120 K if the volume remains constant.

Solution: T₂ = 150K, P₂ = ?, P₁ = 210 mmHg, T₁ = 120K

P₁ / T₁ = P₂ / T₂

P₂ = P₁ . T₂ / T₁

P₂ = 210 x 150 /120

P₂ = 263 mmHg

Example 2: A cylinder contains a gas which has a pressure of 125 mmHg at a temperature of 200 K. Find the temperature of the gas which has a pressure of 100 mmHg if the volume remains constant.

Solution: P1 = 125 mmHg, T1 = 200 K, T2 = ?, P2 = 100 mmHg

P₁ / T₁ = P₂ / T₂

T₂ = T₁ . P₂ / P₁

T₂ = 200 x 100 / 125

T₂ = 160 K

The General Gas Equation

Now we can combine everything we have into one proportion:

The volume of a given amount of gas is proportional to the ratio of its Kelvin temperature and its pressure.
Same as before, a constant can be put in:

PV / T = C

As the pressure goes up, the temperature also goes up, and vice-versa.
Also same as before, initial and final volumes and temperatures under constant pressure can be calculated.

P₁V₁ / T₁ = P₂V₂ / T₂

Example 1: 500 liters of a gas are prepared at 1 atm and 200 °C. The gas is placed into a tank under high pressure. When the tank cools to 20.0 °C, the pressure of the gas is 30 atm. What is the volume of the gas?

Solution: V₁ = 500 L, P₁ = 1 atm, T₁ = 200 ^oC = (200 + 273), T₂ = 20 ^oC = (20 + 273), P₂ = 30 atm, V₂ = ?

P₁V₁ / T₁ = P₂V₂ / T₂

V₂ = P₁V₁T₂ /P₂T₁

V₂ = 1 x 500 x 293 / 30 x 473

V₂ = 10.3 L

Example 2: What is the final volume of a 400 cm³ gas sample that is subjected to a temperature change from 22 °C to 30 °C and a pressure change from 760 mm Hg to 360 mm Hg?

Solution: V₂ = ? V₁ = 400 cm³, T₁ = 22 ^oC = (22 + 273), T₂ = 30 ^oC = (30 + 273), P₁ = 760 mmHg, P₂ = 360 mmHg

P₁V₁ / T₁ = P₂V₂ / T₂

V₂ = P₁V₁T₂ /P₂T₁

V₂ = 760 x 400 x 303 / 360 x 295

V₂ = 867 cm³

Example 3: What is the volume at STP of 720 mL of a gas collected at 20 °C and 3 atm pressure?

Solution: (At STP, T₂ = 273 ^oC, P₂ = 1 atm), V₂ = ?,V₁ = 720 mL, T₁ = 20 ^oC = (20 + 273), P₁ = 3 atm

P₁V₁ / T₁ = P₂V₂ / T₂

V₂ = P₁V₁T₂ /P₂T₁

V₂ = 3 x 720 x 273 / 1 x 293

V₂ = 2013 mL

Dalton’s Law of Partial Pressures

This law states that the total pressure P exerted by a mixture of gases say A, B, C and D is equal to the sum of the partial pressures of each constituent, provided the gases would not chemically react together. In other words, If more than one gas occupy a single container then the number of moles of each gas is in proportional to the pressure of each gas (the gas’ partial pressure) and the total pressure is equal to the sum of all the partial pressures.

P _total = P_A + P_B + P_C + P_D

Where the small P’s are the partial pressure of the individual gases. If a gas is collected over water, it likely to be saturated with water vapour and the total pressures become

P_total = P_gas + P_{water vapour}

P_gas = P_total – P_{water vapour}

The Ideal Gas Law

Ideal gas, or perfect gas, is the theoretical substance that helps establish the relationship of four gas variables, pressure (P), volume(V), the amount of gas(n)and temperature(T). All of these gas laws are based on “ideal” gases. Ideal gases have the following properties:

1. All gas molecules are in motion, and move randomly.
2. Each time the gas particles collide, kinetic energy is conserved (this is called elasticity).
3. The volume of the molecules of gas is negligible (meaning so small it’s not worth stating).
4. Gases do not attract or repel other gas molecules.
5. The kinetic energy of a gas is directly proportional to its temperature (in Kelvin), and is the same for all gases at the same temperature.

Most gases found in nature do not meet all of these qualifications for ideal gas; however, they follow the rules closely enough that we can still use all of the equations above in theory, and it will not present a significant difference from what occurs in nature.

But over a wide range of temperature, pressure, and volume, real gases deviate slightly from ideal.  Since, according to Avogadro, the same volumes of gas contain the same number of moles, chemists could now determine the formulas of gaseous elements and their formula masses.  The idea gas law is:

PV = nRT

Where n is the number of moles of the number of moles and R is a constant called the universal gas constant and is equal to approximately 0.0821 L-atm / mole-K.

Example 1:   At 655mm Hg and 25.0^oC, a sample of Chlorine gas has volume of 750mL. How many moles of Chlorine gas at this condition?

Solution:

 P = 655mm Hg ( 655/760 = 0.862 atm)

 T = (25+273)K

 V = 750mL=0.75L

 n = ?

R = 0.0821 L-atm / mole-K

Solution

n=PV/RT

n = 0.862 x 0.75 / 0.0821 x 298

n = 0.026 mol

Example 2: A sample of butane (C₄H₁₀) of mass 3.728 g is placed in an evacuated bulb of volume 489 mL at 25 ^oC. What is its pressure?

Solution: M = 3.728 g, V = 489 mL = (0.489L), T = 25 ^oC = (25 + 273)K

PV = nRT

P = nRT / V

   = 6.414 x 10^-2 x 0.0821 x 298 / 0.489

   = 3.2 atm