Two identical thermometers made of pyrex glass contain,respectively identical volumes of mercury and methyl alcohol.If the expansion of the glass is taken into account,how many times greater is the distance between the degree marks on the methyl alcohol thermometer than that on the mercury thermometer? (methyl=1200X10^-6(`C), mercury=182X10^-6(`C), glass=9.9X10^-6(`C))
To determine the ratio of the distance between the degree marks on the methyl alcohol thermometer to that on the mercury thermometer, we need to consider the thermal expansion coefficients of the materials involved.
The formula to calculate the change in length due to thermal expansion is given by:
ΔL = L₀ * α * ΔT
Where:
ΔL is the change in length
L₀ is the original length
α is the thermal expansion coefficient
ΔT is the change in temperature
In this case, we want to compare the change in length between the two thermometers, assuming they are subjected to the same temperature change. Since the volume of the mercury and methyl alcohol in the thermometers is identical, the temperature change will also be the same for both.
Let's denote the distance between the degree marks on the methyl alcohol thermometer as L_methyl and on the mercury thermometer as L_mercury.
So, we have:
ΔL_methyl = L_methyl * (α_glass + α_methyl) * ΔT
ΔL_mercury = L_mercury * (α_glass + α_mercury) * ΔT
Here, α_glass is the thermal expansion coefficient of the glass, α_methanol is the thermal expansion coefficient of the methyl alcohol, and α_mercury is the thermal expansion coefficient of the mercury.
Since the thermometers are identical and are subjected to the same temperature change, we can simplify the equation by assuming α_glass is the same for both thermometers:
ΔL_methyl = L_methyl * (α_glass + α_methyl) * ΔT
ΔL_mercury = L_mercury * (α_glass + α_mercury) * ΔT
Now, to find the ratio of the distance between the degree marks on the methyl alcohol thermometer to that on the mercury thermometer, we can divide the two equations:
(ΔL_methyl / ΔL_mercury) = (L_methyl * (α_glass + α_methyl) * ΔT) / (L_mercury * (α_glass + α_mercury) * ΔT)
The ΔT, the temperature change, cancels out, and we are left with:
(ΔL_methyl / ΔL_mercury) = (α_glass + α_methyl) / (α_glass + α_mercury)
Substituting the given values:
(ΔL_methyl / ΔL_mercury) = (9.9e-6 + 1200e-6) / (9.9e-6 + 182e-6)
Calculating this expression will give us the ratio of the distance between the degree marks on the methyl alcohol thermometer to that on the mercury thermometer.
To find the ratio of the distances between the degree marks on the two thermometers, we need to consider the expansion of both the glass and the liquids.
Let's assume the distance between the degree marks on the mercury thermometer is represented by D_m and the distance between the degree marks on the methyl alcohol thermometer is represented by D_a.
The expansion of the glass needs to be accounted for in both thermometers. The expansion coefficient of the glass is given as 9.9 × 10^-6 (°C). So, the expansion of the glass is:
Expansion_glass = Glass coefficient × change in temperature
= 9.9 × 10^-6 (°C) × ΔT
For the mercury thermometer, the total expansion of the glass and mercury needs to be considered. The expansion of the mercury is given as 182 × 10^-6 (°C). Therefore, the expansion of the mercury is:
Expansion_mercury = Mercury coefficient × change in temperature
= 182 × 10^-6 (°C) × ΔT
For the methyl alcohol thermometer, the total expansion of the glass and methyl alcohol needs to be considered. The expansion of the methyl alcohol is given as 1200 × 10^-6 (°C). Therefore, the expansion of the methyl alcohol is:
Expansion_methyl alcohol = Methyl alcohol coefficient × change in temperature
= 1200 × 10^-6 (°C) × ΔT
Given that the two thermometers are made of identical volumes of mercury and methyl alcohol, we can equate the total expansions:
Expansion_glass + Expansion_mercury = Expansion_glass + Expansion_methyl alcohol
9.9 × 10^-6 (°C) × ΔT + 182 × 10^-6 (°C) × ΔT = 9.9 × 10^-6 (°C) × ΔT + 1200 × 10^-6 (°C) × ΔT
Simplifying the equation, we find:
182 × 10^-6 (°C) × ΔT = 1200 × 10^-6 (°C) × ΔT
ΔT cancels out from both sides of the equation. So we're left with:
182 × 10^-6 (°C) = 1200 × 10^-6 (°C)
Now, we can solve for the ratio of the distances between the degree marks:
D_a/D_m = (Expansion_glass + Expansion_methyl alcohol) / (Expansion_glass + Expansion_mercury)
= (9.9 × 10^-6 (°C) + 1200 × 10^-6 (°C)) / (9.9 × 10^-6 (°C) + 182 × 10^-6 (°C))
= 1209.9 × 10^-6 (°C) / 191.9 × 10^-6 (°C)
≈ 6.31
Therefore, the distance between the degree marks on the methyl alcohol thermometer is approximately 6.31 times greater than that on the mercury thermometer.