A sample of neon gas (Ne, molar mass M = 20.2 g/mol) at a temperature of 14.0°C is put into a steel container of mass 43.9 g that’s at a temperature of 47.0°C. The final temperature is ?22.0°C. (No heat is exchanged with the surroundings, and you can neglect any change in the volume of the container.) What is the mass of the sample of neon (in g)?
The Correct Answer and Explanation is :
To find the mass of the neon gas, we need to apply the principle of conservation of energy, which states that the heat lost by the hotter object (steel container) is equal to the heat gained by the colder object (neon gas), assuming no heat is lost to the surroundings.
Step 1: Define Heat Transfer Equation
The heat transfer equation is:
[
Q = mc\Delta T
]
where:
- ( Q ) is the heat energy (in joules),
- ( m ) is the mass of the substance (in grams),
- ( c ) is the specific heat capacity (J/g·°C),
- ( \Delta T ) is the change in temperature (( T_f – T_i )).
Step 2: Heat Lost by the Steel Container
Given:
- Mass of steel: ( m_s = 43.9 ) g
- Specific heat capacity of steel: ( c_s = 0.452 ) J/g·°C
- Initial temperature of steel: ( T_{s,i} = 47.0^\circ C )
- Final temperature: ( T_f = 22.0^\circ C )
[
\Delta T_s = T_f – T_{s,i} = 22.0 – 47.0 = -25.0^\circ C
]
[
Q_s = (43.9 \, \text{g}) (0.452 \, \text{J/g·°C}) (-25.0 \, \text{°C})
]
[
Q_s = -496.85 \text{ J}
]
(The negative sign indicates heat is lost.)
Step 3: Heat Gained by Neon Gas
Given:
- Molar mass of Ne: ( M = 20.2 ) g/mol
- Specific heat capacity of Ne: ( c_{Ne} = 1.03 ) J/g·°C
- Initial temperature of Ne: ( T_{Ne,i} = 14.0^\circ C )
[
\Delta T_{Ne} = T_f – T_{Ne,i} = 22.0 – 14.0 = 8.0^\circ C
]
Heat gained by neon:
[
Q_{Ne} = (m_{Ne}) (1.03 \, \text{J/g·°C}) (8.0 \, \text{°C})
]
Since heat lost by steel is equal to heat gained by neon:
[
496.85 = m_{Ne} (1.03)(8.0)
]
[
m_{Ne} = \frac{496.85}{(1.03 \times 8.0)}
]
[
m_{Ne} = \frac{496.85}{8.24}
]
[
m_{Ne} = 60.3 \text{ g}
]
Final Answer:
The mass of the neon gas is 60.3 g.
Explanation:
The problem involves a thermal equilibrium process where heat energy is transferred from a hotter object (steel container) to a colder one (neon gas) until both reach the same final temperature. By using the heat transfer equation and the principle of conservation of energy, we calculated how much heat the steel container loses and how much heat the neon gains. Solving for the unknown mass of neon, we found that it must be 60.3 g to satisfy energy balance conditions.