Lesson 9: The gaseous State
Video Lesson
Lesson objective
At the end of the lesson, you will be able to :
- Describe the properties of gases using the kinetic molecular theory
- Describe the behavior of gases by using the variables V (volume), T (temperature), P (pressure) and n (number of moles)
- State Boyle’slaw.
- Perform anactivity to show changes in volume and pressure of gases to illustrates Boyle’s law.
- Apply Boyle’slaw in solving problems
- The Gaseous State
- State Charles’law
- Conduct anactivity to toshow changes in volume and temperature of gases to illustrate Charles’ law
- Apply Charles’lawinsolving problems
- UseGay-Lussac’slawinsolvingproblems
- Derive the combined gas law equation from Boyle’s law, Charles’law, and Gay-Lussac’s law
- Use the combined gaslawto calculate changes involume,pressure or temperature
- Define an ideal gas
- Derive an ideal gas equation from Boyle’slaw,Charles’law and Avogadro’s law
- Compare the nature of real gases with ideal gases olive problems related to the ideal gas equation
- Define diffusion.
- Dtate Graham’s law of diffusion
- Carryout an activity to to compare the rate of diffusion of two different gases.
- Apply Graham’slaw of diffusion.
Brainstorming Questions
Consider three water samples that are taken in containers: A, B, and C. The three water samples are kept at different temperatures: 120 ℃, 20 ℃, and -10℃, respectively, and under constant atmospheric pressure.
a. Which sample molecule has the highest kinetic energy?
b. Which sample has the greatest density?
c. Which sample has the most regular arrangement of molecules?
key terms and Concepts
- Gas laws
- Ideal gas equation
- Grahams laws
- Charles Law
Gas laws are a set of fundamental principles that describe the behavior of gases under different conditions of pressure, volume, and temperature. These laws are derived from experimental observations and are crucial for understanding and predicting the properties and behavior of gases. The main gas laws include: Boyles, combined , ideal gad law……
The ideal gas equation, also known as the ideal gas law, is a fundamental equation in the study of gases that relates the pressure ( P ), volume ( V ), temperature ( T ), and amount of substance ( n ) (usually expressed in moles) of an ideal gas. It is expressed mathematically as:[ PV = nRT ]
Graham’s law of effusion describes the relationship between the rate of effusion or diffusion of gases and their molar masses. It states that the rate of effusion or diffusion of a gas is inversely proportional to the square root of its molar mass
Charles’s Law, named after Jacques Charles, describes how gases tend to expand when heated. It states that, at constant pressure, the volume of a given amount of gas is directly proportional to its absolute temperature. In mathematical terms, Charles’s Law can be expressed as T1V2 =T2 V1
3.4. The gaseous State
3.4. 1 The kinetic Molecular Theory of Gases Assumption
- I. Gas particles are in a state of constant, continuous, rapid, random motion and, therefore, possess kinetic energy.
- The motion is constantly interrupted by collisions with molecules or with the container.
- The pressure of a gas is the effect of these molecular impacts.
- II. The volume of the particles is negligible compared to the total volume of the gas.
Gases are composed of separate, tiny invisible particles called molecules. Since these molecules are so far apart, the total volume of the molecules is extremely small compared with the total volume of the gas.Therefore, under ordinary conditions, gas consists of empty space.
This assumption explains why gases are so easily compressed and why they can mix so readily.
III. The attractive forces between the particles are negligible.
There are no forces of attraction or repulsion between gas particles. You can think of an ideal gas molecule as behaving like small billiard balls.
When they collide, they don’t stick together but immediately bounce apart.
IV. The average kinetic energy of gas particles depends on temperature of gases.
At any particular moment, the molecules in a gas have different velocities.
The mathematical formula for kinetic energy is: K.E. = ½ mν2……………where m is mass and ν is velocity of gas molecules.
Because the molecules have different velocities, they have different kinetic energies. However, it is assumed that the average kinetic energy of the molecules is directly proportional to the absolute (Kelvin) temperature of the gas
https://phet.colorado.edu/sims/html/gas-properties/latest/gas-properties_all.html
3.4.2 The Gas Laws
- The gas laws are the products of many experiments on the physical properties of gases, which were carried out over hundreds of years ago.
Simple mathematical equations can be derived that relate a gas’s volume, pressure and temperature.
PV = nRT ……………Ideal Gas Equation
These equations are called state equations because they describe mathematical relationships between the volume, temperature, pressure, and quantity of a gas (number of moles).
1. Boyles law
- Boyle’s law states that the volume of a fixed mass of gas is inversely proportional to the pressure at a constant temperature.
Mathematically given as:
If P1 and V1 represent the initial conditions; and P2 and V2 represent the new or final conditions, Boyle’s law can be written as:
P1V1 = P2V
Example 1
An inflated balloon has a volume of 0.55 L at sea level (1.0 atm) and is allowed to rise to a height of 6.5 km, where the pressure is about 0.40 atm.
Assuming that the temperature remains constant, what is the final volume of the balloon?
Solution:
Givens: Initial conditions Final conditions
P1 = 1.0 atm P2 = 0.40 atm
V1 = 0.55 L V2 = ? V2 = P1V1/P2
V2 = 0.55Lx 1/0.4 = 1.4L
2. Charles Law
- The French physicist, Jacques Charles (1746-1823), was the first person to fill a balloon with hydrogen gas and made the first solo balloon flight.
- Charles investigated quantitative relationship between the volume and temperature of a fixed quantity of gas which is held at constant pressure.
- He stated that the volume of a fixed mass of gas at constant pressure varies directly with the Kelvin temperature.
- In 1848, Lord Kelvin realized that a temperature of -273.15oC is considered as absolute zero. It is theoretically the lowest attainable temperature. Then he set up an absolute temperature scale, or the Kelvin temperature scale, with absolute zero as the starting point on the Kelvin scale.
- For example, Doubling the Kelvin temperature causes the volume of a gas to double, and Reducing the Kelvin temperature by half causes the volume of a gas to decrease by half.
- This relationship between Kelvin temperature and the volume of a gas is known as Charles’ law
V α T at constant presure and constant number of moles(n)
V=KT or V/T=K
V1/T2 = V2/T2
Example
1. A sample of gas at 15 °C and 1 atm has a volume of 2.58 L. What volume will this gas occupy at 38 °C and 1 atm?
Solution:
Given:Initial conditions: T1=15°C = 15+273 =288K, V1 =2.58L
Final Conditions:T2=38°C = 311K, V2=?
By rearranging Charles’ equation: V1/T2 = V2/T2 …………………… V2 can be calculated as:
V2 = V1T2/T1 = (2.58L x 311K )/ 288K = 2.78L
2. A sample of gas has a volume of 2.80L at an unknown temperature. When the sample is submerged in ice water at 0 °C, its volume decreases to 2.57 L. What was its initial temperature (in Kelvin’s and in Celsius)?
Solution:
Given: Initial conditions: T1=? , V1 = 2.80L
Final conditions: T2 = 0°C = 273K, V2=2.57L
Answer: T1 = 277.43K or 24°C
3. Combined Gas Law
- Expresses the relationship between pressure, volume and temperature of a fixed amount of gas.
- A sample of a gas often undergoes changes in temperature ,pressure and volume. When this happens, the three variables must be dea lt with at the same time.
Derivation of the combined gas law:
Boyle’s law: V α 1/P and Charles’ law : VαT then, V T/P ……(combined gas law )
- V = kT/P………………. (where k is a constant) It follows, 𝑃1𝑉1/T1=K and 𝑃2𝑉2/T2 = K
The combined gas law equation is given as follows:
𝑃1𝑉1/T1 = 𝑃2𝑉2/T2
Example : If 50cm3 of gas sample of gas exert a pressure of 60kpa at 35 °C what is the volume at STP?
Solution: 𝑃1𝑉1/T1 = 𝑃2𝑉2/T2 ……………… the combined gas law equation
Given; V1=50cm3, P1=60kpa, T1=35OC = 308K , V2=?
at STP, P2=1atm = 101325pa =101.325kpa, and T2= 0oc = 273K
𝑉2= 𝑃1𝑉1 T2 /𝑃2T1 = (60kpa x50cm3 x 273K)/(101.325kpa x308K)
V2 = 26.243cm3
4. Avogadro’s law:
states that the volume of the gas is directly proportional to the number of mole of gas, when the temperature and pressure are held constant.
Mathematically, V α n; where V is the volume and n is number of moles.
Common example of Avogadro’s law’s is the deflation of automobile tires.
5. The Ideal Gas Equation
- Hypothetical gas that obeys the gas laws. Real gases only obey the ideal gas laws closely at high temperature and low pressure . The ideal gas lawis a combination of:
- Boyle’s law………….. V α 1/p
- Charles’ law ………….V = k1/p Vα nT
- Avogadro’s law…………….Vα n
combining the three laws ……………. V α (nT) 1/p
PV/nT= R…………. where ,R is the ideal gas constant and R= 0.082 L.atm. mol-1 oK-1 at STP.
Thus, PV =nRT………….the ideal gas equation.
6. Grahams Law
Graham’s Law of diffusion states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, it can be expressed as: $\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}$
where:
- M1 and M2 are the molar masses of gas 1 and gas 2, respectively.
- r1 and r2 are the rates of diffusion of gas 1 and gas 2, respectively.
This means that lighter gases diffuse faster than heavier gases. For example, if you have hydrogen (with a low molar mass) and oxygen (with a higher molar mass), hydrogen will diffuse more quickly than oxygen.
Graham’s Law is useful in understanding the behavior of gases in various scientific and industrial applications, including effusion and gas mixtures.
https://phet.colorado.edu/sims/html/diffusion/latest/diffusion_all.html