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How much heat, in joules, must be added to 0.250 mol of Ar(g) to raise its temperature from 20°C to 36°C at constant pressure?
A. 50.0
B. 83.2
C. 187
D. 200

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Last updated date: 20th Jun 2024
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Answer
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Hint: Here, the temperature of argon gas needs to be raised from 20°C to 36°C at constant pressure by adding a certain amount of heat energy. Argon is a monatomic gas. Here, we have to find out the amount of heat required to increase the temperature of argon gas that means argon will absorb some amount of heat which will lead to the increase in temperature of that gas. We have to calculate the amount of heat absorbed by argon gas.

Formula used: $q = n \times {C_p} \times \Delta T$
$q = $ Amount of heat required to be added in 0.250 mol Ar(g) to raise its temperature that is heat capacity
$n = $ Number of moles of Ar(g)
$\Delta T = $ Rise in temperature
${C_p} = $ Molar heat capacity at constant pressure

Complete step by step answer:
The certain amount of heat needs to be added in order to raise the temperature of argon gas. As we are aware, the temperature is nothing but the measure of the total kinetic energy of the particles of a particular object. Therefore, the initial temperature of Ar(g) is because of the kinetic energy of the Argon atoms in the gas. When we are adding more heat in the gas, the gas will absorb heat which will then contribute to kinetic energy of Ar atoms in the gas and will raise the temperature of the gas. Thus, we can say that the heat added will be proportional to that of change in temperature.
We have to calculate the amount of heat added that is heat capacity and for that we will be using the following formula.
$q = n \times {C_p} \times \Delta T$
Heat capacity of a substance is expressed as the ratio of amount of heat absorbed by a substance to that of the change in temperature of that substance.
Let us first note down the quantities given in order to proceed with calculations.
Number of moles of Ar (n) = 0.250mol
Initial temperature, ${T_1} = 20^\circ C$
Final temperature, ${T_2} = 36^\circ C$
Step 1: The first step is to calculate $\Delta T$, the difference between initial and final temperature of Ar(g).
 We can calculate this by subtracting the initial temperature of Ar from the final temperature of Ar.
$\therefore \Delta T = {T_2} - {T_1}$
On putting the values in above formula, we get,
$\therefore \Delta T = (36 - 20)^\circ C = 16^\circ C$
Step 2: Now, we know the values of n and $\Delta T$, we need to find out the value for ${C_p}$. Here, ${C_p}$ is the molar heat capacity at constant pressure.
Let us first understand the meaning of heat capacity of the substance. The molar heat capacity signifies the amount of heat required for increasing temperature of substance by one degree. For a gaseous substance, heat capacity is expressed as the amount of heat required for 1 mole of gas to raise its temperature through 1 kelvin.
Now, ${C_p}$, molar heat capacity at constant pressure is the amount of heat energy absorbed or evolved by an unit mass of a substance accompanied by change in its temperature under constant pressure.
Here, the value for ${C_p}$ is not provided. But, we know that Argon is a monatomic gas.
As derived by Meyer’s relation,
${C_p} = \dfrac{5}{2}R$…………… (1)
At constant pressure, heat capacity is given by,
$q = n \times {C_p} \times \Delta T$
On putting the value of ${C_p}$ from relation given in formula (1), we get,
$q = n \times \dfrac{5}{2}R \times \Delta T$…………….. (2)
Here, R is a gas constant.
We can use the formula (2) to obtain the value of the amount of heat required to raise temperature of Ar.
Step 3: We know, $\Delta T = 16^\circ C$ and $R = 8.314J/mol.K$
 On substituting respective values in formula (2),
$q = 0.250 \times \frac{5}{2} \times 8.314 \times 16$
$\therefore q = 83.14 \approx 83.2joules$

So, the correct answer is B.

Additional information:
In step 2 of the solution, we have mentioned Mayer’s relation. Mayer’s relation or Mayer’s law states the relation between molar heat capacity of an ideal gas at constant pressure (${C_p}$) and molar heat capacity of an ideal gas at constant volume (${C_v}$). According to Mayer’s relation,
${C_p} - {C_v} = R$
Here, R is a gas constant and it has value equals to 8.314J/mol.K
Mayer's relation implies that the molar heat capacity of an ideal gas at constant pressure is greater than that of molar heat capacity of an ideal gas at constant volume.

Note: As stated in solution, heat capacity of argon is expressed as the amount of heat required to raise temperature for 1 mole of argon gas through 1 kelvin. In question, the unit of temperature used is degree Celsius, not kelvin. Still, in the first step of solution while calculating $\Delta T$ we have used temperature in degree Celsius only, not in kelvin.
This is because an increase in the scale of degree Celsius is equivalent to that of an increase in Kelvin scale. This implies that, if we raise the temperature of water through 1°C, then it will be equal to that of the increase in temperature of water through 1K.