Answer :
The equilibrium temperature, calculated using the principle of conservation of energy, is approximately 50 degrees Celsius. So, the correct choice is option d.
To find the equilibrium temperature of the mixture, we can use the principle of conservation of energy, which states that the total heat gained by one substance equals the total heat lost by the other.
The equation for heat transfer is given by [tex]\( Q = mc\Delta T \)[/tex], where:
- [tex]\( Q \)[/tex] is the heat transferred,
- [tex]\( m \)[/tex] is the mass,
- [tex]\( c \)[/tex] is the specific heat capacity,
- [tex]\( \Delta T \)[/tex] is the change in temperature.
1. Calculate heat gained and lost:
For nitrogen: [tex]\( Q_{\text{N}_2} = m_{\text{N}_2}c_{\text{N}_2}\Delta T_{\text{N}_2} \)[/tex]
For helium: [tex]\( Q_{\text{He}} = m_{\text{He}}c_{\text{He}}\Delta T_{\text{He}} \)[/tex]
2. Apply conservation of energy:
[tex]\( Q_{\text{N}_2} = -Q_{\text{He}} \)[/tex]
3. Substitute and solve for [tex]\( \Delta T \)[/tex]:
[tex]\[ m_{\text{N}_2}c_{\text{N}_2}\Delta T_{\text{N}_2} = -m_{\text{He}}c_{\text{He}}\Delta T_{\text{He}} \][/tex]
4. Calculate equilibrium temperature:
\[tex][ \text{Equilibrium Temperature} = \text{Initial temperature of helium} + \Delta T_{\text{He}} \][/tex]
Using the specific heat capacities of nitrogen and helium, and their initial temperatures, you can find [tex]\( \Delta T_{\text{He}} \)[/tex] and then the equilibrium temperature. Therefore, option d is the suitable choice.
