Sure, let's solve this question step by step:
The general formula for heat required ([tex]\(Q\)[/tex]) to increase the temperature of a substance is given by:
[tex]\[ Q = n \cdot c \cdot \Delta T \][/tex]
where:
- [tex]\(n\)[/tex] is the number of moles of the substance
- [tex]\(c\)[/tex] is the specific heat capacity (at constant volume or constant pressure as specified)
- [tex]\(\Delta T\)[/tex] is the change in temperature
Let's determine the heat required for each case:
### a) Helium (He) at constant volume:
Given:
- Number of moles ([tex]\(n\)[/tex]) = 7
- Temperature increase ([tex]\(\Delta T\)[/tex]) = 50 K
- Specific heat capacity at constant volume for Helium ([tex]\(c_v\)[/tex]) = 12.5 J/mol.K
The heat required is:
[tex]\[ Q_{He} = n \cdot c_v \cdot \Delta T \][/tex]
[tex]\[ Q_{He} = 7 \cdot 12.5 \cdot 50 \][/tex]
[tex]\[ Q_{He} = 4375 \, \text{J} \][/tex]
### b) Argon (Ar) at constant pressure:
Given:
- Number of moles ([tex]\(n\)[/tex]) = 7
- Temperature increase ([tex]\(\Delta T\)[/tex]) = 50 K
- Specific heat capacity at constant pressure for Argon ([tex]\(c_p\)[/tex]) = 20.8 J/mol.K
The heat required is:
[tex]\[ Q_{Ar} = n \cdot c_p \cdot \Delta T \][/tex]
[tex]\[ Q_{Ar} = 7 \cdot 20.8 \cdot 50 \][/tex]
[tex]\[ Q_{Ar} = 7280 \, \text{J} \][/tex]
### c) Nitrogen ([tex]\(N_2\)[/tex]) at constant volume:
Given:
- Number of moles ([tex]\(n\)[/tex]) = 7
- Temperature increase ([tex]\(\Delta T\)[/tex]) = 50 K
- Specific heat capacity at constant volume for Nitrogen ([tex]\(c_v\)[/tex]) = 20.8 J/mol.K
The heat required is:
[tex]\[ Q_{N2} = n \cdot c_v \cdot \Delta T \][/tex]
[tex]\[ Q_{N2} = 7 \cdot 20.8 \cdot 50 \][/tex]
[tex]\[ Q_{N2} = 7280 \, \text{J} \][/tex]
### d) Carbon Dioxide ([tex]\(CO_2\)[/tex]) at constant pressure:
Given:
- Number of moles ([tex]\(n\)[/tex]) = 7
- Temperature increase ([tex]\(\Delta T\)[/tex]) = 50 K
- Specific heat capacity at constant pressure for Carbon Dioxide ([tex]\(c_p\)[/tex]) = 37.1 J/mol.K
The heat required is:
[tex]\[ Q_{CO2} = n \cdot c_p \cdot \Delta T \][/tex]
[tex]\[ Q_{CO2} = 7 \cdot 37.1 \cdot 50 \][/tex]
[tex]\[ Q_{CO2} = 12985 \, \text{J} \][/tex]
### Summary:
- Heat required for Helium (He) at constant volume: 4375 J
- Heat required for Argon (Ar) at constant pressure: 7280 J
- Heat required for Nitrogen ([tex]\(N_2\)[/tex]) at constant volume: 7280 J
- Heat required for Carbon Dioxide ([tex]\(CO_2\)[/tex]) at constant pressure: 12985 J
I hope this step-by-step solution helps! If you have further questions, feel free to ask.
