To find the new temperature of the air in the glass sphere after it has been heated, we can use the principles of gas laws and the relationship between pressure, volume, and temperature. In this scenario, we have a fixed volume of air that expands as it is heated, causing the mercury level in the pipe to rise. Let's break down the problem step by step.

Understanding the Initial Conditions

The initial volume of the air in the sphere is given as 7 liters, which we can convert to cubic meters for consistency in SI units:

• 7 liters = 0.007 m³

The initial temperature of the air is 300 K. The pressure exerted by the air in the sphere can be calculated using the ideal gas law, which states:

PV = nRT

Where:

• P = pressure
• V = volume
• n = number of moles of gas
• R = ideal gas constant (8.314 J/(mol·K))
• T = temperature in Kelvin

Analyzing the Change in Mercury Level

When the air is heated, it expands, and this expansion causes the mercury level in the right arm of the pipe to rise by 5 mm. To find the volume of mercury displaced, we can calculate the volume corresponding to this height change:

The cross-sectional area of the pipe is given as 10 cm², which is equivalent to:

• 10 cm² = 0.001 m²

The volume of mercury displaced (V_displaced) can be calculated as:

V_displaced = Area × Height

Substituting the values:

V_displaced = 0.001 m² × 0.005 m = 0.000005 m³

Calculating the New Volume of Air

The new volume of air (V_new) in the sphere after the mercury displacement is:

V_new = V_initial + V_displaced

Thus:

V_new = 0.007 m³ + 0.000005 m³ = 0.007005 m³

Applying the Ideal Gas Law

Since the number of moles of air (n) remains constant, we can set up the relationship between the initial and final states of the gas using the ideal gas law:

P_initial × V_initial / T_initial = P_final × V_new / T_final

Assuming the pressure change due to the mercury displacement is negligible, we can simplify this to:

V_initial / T_initial = V_new / T_final

Rearranging gives us:

T_final = T_initial × (V_new / V_initial)

Substituting the Values

Now we can substitute the known values:

T_final = 300 K × (0.007005 m³ / 0.007 m³)

Calculating this gives:

T_final = 300 K × 1.0007142857 ≈ 300.214 K

Final Result

The new temperature of the air in the glass sphere after heating is approximately:

T_final ≈ 300.21 K

This result shows that the temperature of the air increased slightly due to the expansion caused by heating, which is consistent with the behavior of gases as described by the ideal gas law.