Charles’ Law: V1/T1 = V2/T2, what is it, where to use it?

The formula V₁/T₁ = V₂/T₂ represents Charles’ law, a direct proportional relationship between the temperature and volume of a gas at a constant pressure. It is also known as the law of volumes.

The behavior of mercury in household thermometers is an important application of Charles’ law in everyday life. At a high temperature, liquid mercury expands and rises above the narrow tube of the thermometer. Contrarily, it contracts and falls down as the temperature drops.

Similarly, Charles’ law can be applied to gases. Let us find out how through this article.

What is Charles’ law of gases? – V₁/T₁ = V₂/T₂?

As per the kinetic theory of gases, in an ideal gas sample, the individual gas molecules are continuously moving, in random directions, occupying maximum space.

If the pressure and concentration of the gas are kept constant, upon increasing its temperature, the gas molecules expand and occupy more space, i.e., a greater volume, and vice versa.

Thus, according to Charles’ law, the volume of a gas is directly proportional to the Kelvin temperature. Mathematically represented as:

V α T

Converting proportionality into equality gives us:

V = k T

k = V/T

• Where V = volume, T = temperature and k = proportionality constant

If the temperature of a gaseous system is changed from T₁ to T₂, its volume changes from V₁ to V₂, and the relationship between the two variables can be expressed as per Charles’ law:

V₁/T₁ = k……Equation (i)

V₂/T₂ = k ….Equation (ii)

Combining equations (i) and (ii), the effect of varying temperature conditions can be studied on the volume of a gas via equation (iii).

V₁/T₁ = V₂/T₂……..Equation (iii)

What do the components of V₁/T₁ = V₂/T₂ represent?

For a gaseous system undergoing some change in temperature and volume at a constant pressure and concentration, V₁/T₁ = V₂/T₂ represents:

• V₁ = Initial volume of the gas (Units = L or m³)
• T₁ = Initial temperature (Units = K)
• V₂ = Final volume of the gas (Units = L or m³)
• T₂ = Final temperature (Units = K)

Where and how to use V₁/T₁ = V₂/T₂? – Examples

The formula V₁/T₁ = V₂/T₂ can be used to find the initial or final volume and/or temperature of a gas by rearranging this equation into the following four different ways, depending upon which variable is unknown.

Another important point is that consistency in units is very important. The units of volume (V₁ and V₂) should be the same on either side of V₁/T₁ = V₂/T₂. However, temperature (T₁ and T₂) should always be given Kelvin.

The temperature in °C can be converted into K as follows:

1 K = 1°C + 273

For example, V₁ liters of hydrogen (H₂) gas at 27°C is held in a piston with constant pressure. As the temperature is raised to 327 °C, the volume of the gas increases to 20 L. Find V₁.

Solution

The data obtained from the question statement is:

V₁ = ?

T₁ = 27° C

V₂ = 20 L

T₂ = 327°C

As a first step, we need to convert both T₁ and T₂ from °C to K (absolute scale).

1 K = 1°C + 273

T₁ = 27 + 273 = 300 K

T₂ = 327 + 273 = 600 K

Now let us plug in all the given data into V₁/T₁ = V₂/T₂:

Cross-multiply and make V₁ the subject of the formula as shown below:

V₁ x 600 = 20 x 300

600 V₁  = 6000

V₁ = 6000/600

∴ V₁ = 10 mL

Result: The initial volume of hydrogen gas in this example is 10 mL.

Another example is- A scientist has a gas chamber with a constant pressure of 186 mm Hg. The volume of the chamber is 100 mL at a temperature of 200 K. However, if the temperature is raised by 50 K, its volume increases to V₂. Find the value of V₂ using the formula V₁/T₁ = V₂/T₂.

Solution

The data obtained from the question statement is:

V₁ = 100 mL

T₁ = 200 K

V₂ = ?

T₂ = 200 + 50 = 250 K (as the temperature is raised by 50 K)

As the pressure inside the gas chamber is kept constant, so we can apply Charles’ law to find the final volume of the gas chamber i.e., V₂.

Cross-multiply and make V₂ the subject of the formula as shown below:

100 x 250 = 200 x V₂

25000 = 200 V₂

V₂ = 25000/200

∴ V₂ = 125 mL

Result: The final volume of the gas chamber is 125 mL.

As per Charles’ law, increasing the temperature of the gas chamber increases the volume of gas.

More examples on V₁/T₁ = V₂/T₂

A car tire is filled with about 70 L of air on a summer day when the temperature is a very warm 50°C. What is the temperature on a very cold winter day when the volume decreases to 56 L?

As per the question statement; V1 = 70 L T1 = 50°C V2 = 56 L T2 = ? Converting temperature from °C to K: 1 K = 1°C + 273 T1 = 50 + 273 = 323 K Substituting the above data into the formula: Making T2 the subject of the formula by cross multiplication: 70 x T2 = 56 x 323 70 T2 = 18088 T2 = 18088/70 T2 = 258.4 K As T1 was given in °C in the question statement, therefore we need to convert T2 from K to °C: 1 °C = 1 K – 273 T2 = 258.4 – 273 ∴ T2 = -14.6°C Result: The temperature on the very cold winter day was -14.6° C.

At what temperature the gas occupies a volume of 0.3 L if the volume of the gas is 0.00065 m3 at 80°C?

As per the question statement; V1 = 0.3 L T1 = ? V2 = 0.00065 m3 T2 = 80°C To bring consistency in units, we need to convert both V1 and V2 into the same units. 1 m3 = 1000 L V2 = 0.00065 x 1000 = 0.65 L In contrast, we need to convert T2 from °C to Kelvin (K) as shown below: 1 K = 1°C + 273 T2 = 80 + 273 T2 = 353 K Substituting all the data into the formula:   Making T1 the subject of the formula via cross-multiplication: 0.3 x 353 = 0.65 x T1 105.9 = 0.65 T1 T1 = 105.9/0.65 T1 = 162.9 K 1 °C = 1 K – 273 T1 = 162.9 – 273 ∴ T1 = -110.1°C Result: The temperature of the gas is -110.1°C when it occupies a volume of 0.3 L, as per this example.

FAQ

What is Charles’ law?

Charles’ law of gases states: At a constant pressure (P), for a fixed amount of gas (n), the volume (V) of the gas is directly proportional to the absolute temperature (T). V α T V = kT

What is represented by V1/T1 = V2/T2?

At a constant pressure and concentration, for an ideal gas undergoing temperature change from T1 to T2, the volume of the gas changes from V1 to V2. Hence V1/T1 = V2/T2 represents: V1= initial volume of gas T1 = initial gas temperature V2 = final volume of gas T2 = final gas temperature

What are the units of V and T in V1/T1 = V2/T2?

In V1/T1 = V2/T2, the volume (V) is measured in liters (L) or meter cubic (m3) where 1 m3 = 1000 L. Temperature (T) must always be taken in the absolute scale, i.e., Kelvin (K).

Draw a straight-line graph to show the relationship between volume and temperature as per Charles’ law.

The graph shown below is known as an isobar. At a constant pressure, volume is directly proportional to the temperature of a gas. A straight-line graph is obtained with volume (dependent variable) on the y-axis versus temperature (independent variable) on the x-axis. Increasing the pressure (P1, P2, P3) decreases the gradient of the graph. However, each straight line is specific to a constant pressure value.

As per Charles law, V = kT where V= volume, T= temperature, and k = constant. Find the value of proportionality constant k for a gas having a volume of 75 liters at 27°C. What are the units of k?

V= kT…..Charles’ law Making k the subject of the formula gives us: k = V/T Substituting the known values: k = 75/(27 + 273) k = 0.25 The units of k are L/K. So the value of k for this particular gas is 0.25 L/K.