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Chapter 13: Q82GP (page 359)

The first real length standard, adopted more than 200 years ago, was a platinum bar with two very fine marks separated by what was defined to be exactly one meter. If this standard bar was to be accurate to within \( \pm 1.0\;{\rm{\mu m}}\), how carefully would the trustees have needed to control the temperature? The coefficient of linear expansion is \(9 \times {10^{ - 6}}\;{\rm{/^\circ C}}\) .

Short Answer

Expert verified

The required temperature is\(\Delta T = \pm 0.11{\rm{^\circ C}}\).

Step by step solution

01

Given data

The standard bar should be within\(\Delta L = \pm 1.0\;{\rm{\mu m}}\).

The coefficient of linear expansion is\(\alpha = 9 \times {10^{ - 6}}\;{\rm{/^\circ C}}\).

The length is \(L = 1\;{\rm{m}}\).

02

Expression for linear expansion

In this problem, use the expression for the linear expansion of a material to determine the temperature required.

03

Calculation of the temperature

The relation to find the requiredtemperature can be written as:

\(\Delta L = \alpha L\Delta T\)

Here,\(\Delta T\)is the required temperature.

On plugging the values in the above relation, you get:

\(\begin{aligned}\left( {1\;{\rm{\mu m}} \times \frac{{{{10}^{ - 6}}\;{\rm{m}}}}{{1\;{\rm{\mu m}}}}} \right) &= \left( {9 \times {{10}^{ - 6}}\;{\rm{/^\circ C}}} \right)\left( {1\;{\rm{m}}} \right)\Delta T\\\Delta T &= \pm 0.11{\rm{^\circ C}}\end{aligned}\)

Thus, \(\Delta T = \pm 0.11{\rm{^\circ C}}\)is the required temperature.

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Most popular questions from this chapter

Question: (I)\({\rm{C}}{{\rm{O}}_{\rm{2}}}\)exists in what phase when the pressure is 35 atm and the temperature is 35°C (Fig. 13–23)?

Question: (II)Show that for a mixture of two gases at the same temperature, the ratio of their rms speeds is equal to the inverse ratio of the square roots of their molecular masses,\(\frac{{{{\bf{v}}_{\bf{1}}}}}{{{{\bf{v}}_{\bf{2}}}}}{\bf{ = }}\sqrt {\frac{{{{\bf{M}}_{\bf{2}}}}}{{{{\bf{M}}_{\bf{1}}}}}} \).

Question: (II)If the pressure in a gas is tripled while its volume is held constant, by what factor does\({v_{{\rm{rms}}}}\)change?

Question: (II)Two isotopes of uranium,\({}^{{\bf{235}}}{\bf{U}}\)and\({}^{{\bf{238}}}{\bf{U}}\)(the superscripts refer to their atomic masses), can be separated by a gas diffusion process by combining them with fluorine to make the gaseous compound\({\bf{U}}{{\bf{F}}_{\bf{6}}}\). Calculate the ratio of the rms speeds of these molecules for the two isotopes, at constant T. Use Appendix B for masses.

Question: A flat bimetallic strip consists of a strip of aluminum riveted to a strip of iron. When heated, the strip will bend. Which metal will be on the outside of the curve? Why? (Hint: See Table 13–1.)
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