Question

A gas at $100°C$fills volume $V_0$. If the pressure is held constant, what is the volume if

(a) the Celsius temperature is doubled and

(b) the Kelvin temperature is doubled?

Short answer

(a) The volume if the Celsius temperature is doubled is $1.27V_0$

(b)The volume if the Kelvin temperature is doubled is$2V_0$

Step-by-step solution

Defining Ideal Gas law

Theideal gas law shows the relationship of the four state variables for an ideal gas, namely the volume$V$of the container, the pressure $p$exerted by the gas, temperature $T$of the gas, and the number of moles $n$of the gas in the container.

The ideal gas law is given by $PV=nRT$

If the pressure is constant for the same gas, we get the relation for two points of time by

$$\begin{gathered} \frac{V_2}{V_1}=\frac{T_2}{T_1} \\ \Rightarrow V_2=\left(\frac{T_2}{T_1}\right)V_1 \end{gathered}$$

Finding the volume if the Celsius temperature is doubled 

(a) Here $V₁=V$. The initial temperature in Celsius is $T₁=100°C$. This temperature is doubled and is$T₂=200°C$.
We convert these two temperature values to kelvins and substitute these values into the equation $V_2=\left(\frac{T_2}{T_1}\right)V_1$to get

$$\begin{gathered} T₁=T_C+273=100°C+273=373K \\ T₂=T_C+273=200°C+273=473K \end{gathered}$$

Now we obtain the value of$V₂$when the Celsius temperature is doubled

role="math" localid="1648399030485" $$\begin{gathered} V_2=\left(\frac{T_2}{T_1}\right)V_1 \\ \Rightarrow V_2=\left(\frac{473}{373}\right)V_0 \\ =\mathbf{1}\mathbf{.}\mathbf{27}{\mathit{V}}_{\mathbf{0}} \end{gathered}$$

Finding the volume if the Kelvin temperature is doubled 

(b) The initial temperature in Kelvin is $T₁=373K$. This value is doubled and is$746K$
Now we obtain the value of $V₂$if the Kelvin temperature is doubled

$$\begin{gathered} V_2=\left(\frac{T_2}{T_1}\right)V_1 \\ =\left(\frac{746}{373}\right)V_0 \\ =\mathbf{2}{\mathit{V}}_{\mathbf{0}} \end{gathered}$$