Answer :
31.5 Celsius is the final temperature of the mixture. Thus the correct answer is 31.5 Celsius (option C).
To find the temperature of the mixture, we need to use the principle of energy conservation, which states that the total internal energy of the system remains constant. We'll calculate the heat absorbed or released by each gas and then equate the sum of their energies to zero, as the net change in internal energy is zero.
Let's denote:
- [tex]\( m_{\text{CO}_2} = 22 \)[/tex] g (mass of CO₂)
-[tex]\( m_{\text{O}_2} = 16 \)[/tex] g (mass of O₂)
- [tex]\( c_{\text{CO}_2} = c_{\text{O}_2} = c \)[/tex] (assuming specific heat capacity is the same for both gases)
- [tex]\( T_{\text{CO}_2} = 27 \)[/tex] °C (initial temperature of CO₂)
- [tex]\( T_{\text{O}_2} = 37 \)[/tex]°C (initial temperature of O₂)
- [tex]\( T_{\text{final}} \)[/tex] (final temperature of the mixture)
The heat absorbed or released by each gas can be calculated using the equation [tex]\( Q = mc\Delta T \),[/tex] where ( Q ) is the heat transferred, ( m ) is the mass, ( c ) is the specific heat capacity, and [tex]\( \Delta T \)[/tex] is the change in temperature.
For CO₂:
[tex]\[ Q_{\text{CO}_2} = m_{\text{CO}_2}c\Delta T_{\text{CO}_2} \][/tex]
For O₂:
[tex]\[ Q_{\text{O}_2} = m_{\text{O}_2}c\Delta T_{\text{O}_2} \][/tex]
According to the principle of energy conservation,
[tex]\( Q_{\text{CO}_2} + Q_{\text{O}_2} = 0 \).[/tex]
Let's first calculate [tex]\( \Delta T_{\text{CO}_2} \)[/tex] and [tex]\( \Delta T_{\text{O}_2} \):[/tex]
[tex]\[ Q_{\text{CO}_2} = 22c(T_{\text{final}} - 27) \][/tex]
[tex]\[ Q_{\text{O}_2} = 16c(T_{\text{final}} - 37) \][/tex]
Since [tex]\( Q_{\text{CO}_2} + Q_{\text{O}_2} = 0 \):[/tex]
[tex]\[ 22c(T_{\text{final}} - 27) + 16c(T_{\text{final}} - 37) = 0 \][/tex]
[tex]\[ 22(T_{\text{final}} - 27) + 16(T_{\text{final}} - 37) = 0 \][/tex]
[tex]\[ 22T_{\text{final}} - 594 + 16T_{\text{final}} - 592 = 0 \][/tex]
[tex]\[ 38T_{\text{final}} = 1186 \][/tex]
[tex]\[ T_{\text{final}} = \frac{1186}{38} \][/tex]
[tex]\[ T_{\text{final}} = 31.21 \][/tex]
Therefore, the final temperature of the mixture is approximately ( 31.21 ) °C. Since this value is closest to ( 31.5 ) °C, the correct answer is option C. Thus the correct answer is 31.5 Celsius (option C).
