Question

1/2 mole of helium is contained in a container at STP. How much heat energy is needed to double the pressure of the gas (volume is constant) heat capacity of has is 3J/g/K
(A) 1436
(B) 736
(C) 1638
(D) 5698

Answer 1

Hint: The pressure of a given mass of gas is directly proportional to the absolute temperature provided that the volume is kept constant. Use that relation to calculate the temperature difference of the two different states of the system (the high pressure state and the stp pressure state). Standard temperature is considered as zero degree Celsius or 273 Kelvin.
Formula used: In this solution we will be using the following formulae;
\[P = kT\] where \[P\] is the pressure of an ideal gas at a certain state, \[T\] and is the absolute temperature of the gas at the same state, \[k\] is a proportionality constant.
\[Q = nM{c_v}\Delta T\] where \[Q\] is the heat absorbed by a gas,\[M\] is the molar mass,\[n\] is the number of moles of the gas, \[{c_v}\] is the specific heat capacity at constant volume and \[\Delta T\] is the difference in temperature after a particular amount of absorbed heat.

Complete Step-by-Step solution:
To calculate for heat needed, we must first calculate the temperature necessary for the condition to hold.
Generally, it is known that the pressure of a gas is directly proportional to temperature of the gas provided the volume is held constant. Hence
\[P = kT\] where \[P\] is the pressure of an ideal gas at a certain state, \[T\] and is the absolute temperature of the gas at the same state, \[k\] is a proportionality constant.
Hence, for the first state, we have
\[{P_1} = k{T_1} = 273k\] (since the standard temperature is 373 K)
\[ \Rightarrow k = \dfrac{{{P_1}}}{{273}}\]
For the second state, we have
\[ \Rightarrow 2{P_1} = k{T_2}\]
Inserting the known \[k\] expression, we have
\[2{P_1} = \left( {\dfrac{{{P_1}}}{{273}}} \right){T_2}\]
\[ \Rightarrow 2 = \dfrac{{{T_2}}}{{273}}\]
Hence, we have
\[{T_2} = 2 \times 273 = 546K\]
Now, the heat required can be given by
\[Q = nM{c_v}\Delta T\] where \[n\] is the number of moles of the gas,\[M\] is the molar mass,\[{c_v}\] is the specific heat capacity at constant volume and \[\Delta T\] is the difference in temperature after a particular amount of absorbed heat.
Hence, we have
\[Q = \left( {0.5} \right)4\left( 3 \right)\left( {546 - 273} \right)\]
By computation,
\[Q = 1638J\]

The correct option is C

Note: Alternatively, instead for calculating for \[k\] and then using its value in the second state, we could simply divide state 1 by state 2, and hence, we have
\[\dfrac{{{P_1}}}{{{P_2}}} = \dfrac{{{T_1}}}{{{T_2}}}\]
Then by making \[{T_2}\] subject of formula, we have
\[{T_2} = \dfrac{{{P_2}{T_1}}}{{{P_1}}}\]
Hence, by inserting the known values and expressions, we have
\[{T_2} = \dfrac{{2{P_1}\left( {273} \right)}}{{{P_1}}} = 546K\]
Which is identical to what is gotten above.