he column of mercury in a barometer (see Figure below) has a height of 0.760 m when the pressure is one atmosphere and the temperature is 0.0 °C. Ignoring any change in the glass containing the mercury, what will be the height of the mercury column for the same one atmosphere of pressure when the temperature rises to 36.6 °C on a hot day? Hint: The pressure in the barometer is Pressure = ρgh, and the density ρ changes when the temperature changes.

ans is right.

Volume thermal expansion coef Hg 0.000181/K

∆V = αᵥV∆T, αᵥ is volume thermal expansion coef

∆V = αᵥV∆T

∆V/V = αᵥ∆T,

because cross sectional area is fixed

∆V/V = ∆h/h = αᵥ∆T,

∆h = hαᵥ∆T, = (760mm)(0.000181/K)(36.6K) = 5.03 mm

new h = (760+5.03) = 765.03mm

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To find the height of the mercury column when the temperature rises to 36.6 °C, we need to take into account the change in density of mercury.

The pressure in the barometer is given by the equation Pressure = ρgh, where ρ is the density of the mercury, g is the acceleration due to gravity, and h is the height of the mercury column. Since the pressure is constant at one atmosphere, we can set up the equation:

Pressure1 = Pressure2

Where,
Pressure1 = ρ1 * g * h1 (at 0.0 °C)
Pressure2 = ρ2 * g * h2 (at 36.6 °C)

We know that the pressure is the same in both cases, so we can set up the equation as:

ρ1 * g * h1 = ρ2 * g * h2

Now, let's examine how the density ρ changes with temperature. The density of a substance changes with temperature according to the equation:

ρ2 = ρ1 * (1 + β * ΔT)

Where,
ρ1 = density at 0.0 °C (known)
ρ2 = density at 36.6 °C (unknown)
β = coefficient of volume expansion of mercury (known)
ΔT = change in temperature in Celsius (36.6 °C - 0.0 °C = 36.6 °C)

Given that the coefficient of volume expansion of mercury (β) is approximately 0.000182°C⁻¹, we can substitute these values into the equation:

ρ2 = ρ1 * (1 + 0.000182 * ΔT)

Now, we can substitute this value of ρ2 into our earlier equation:

ρ1 * g * h1 = ρ2 * g * h2

ρ1 * g * h1 = ρ1 * (1 + 0.000182 * ΔT) * g * h2

ρ1 * h1 = (1 + 0.000182 * ΔT) * ρ1 * h2

Since ρ1 * h1 = ρ24 * h2:

h2 = h1 / (1 + 0.000182 * ΔT)

Now we can calculate the height of the mercury column (h2) for the same one atmosphere of pressure at 36.6 °C using the given information:

h1 = 0.760 m
ΔT = 36.6 °C

Substituting these values into the equation:

h2 = 0.760 m / (1 + 0.000182 * 36.6 °C)

Simplifying the equation, we get:

h2 ≈ 0.760 m / (1 + 0.0066632)

h2 ≈ 0.760 m / 1.0066632

h2 ≈ 0.7551 m

Therefore, the height of the mercury column for the same one atmosphere of pressure when the temperature rises to 36.6 °C will be approximately 0.7551 meters.