Re: Is there any way of stopping the thermal expansion of mercury.

The short answer is no, thermal expansion cannot be contained by putting something inside another material. Thermal expansion is caused by the change in atom spacing as the temperature changes, and all materials will change some. Fused quartz is very close to zero thermal expansion over a wide range, but even it is not perfectly zero, and the forces generated by anything contained within it would eventually shatter the quartz.

Thermal expansion is a major concern for large structures such as bridges. Railroads suffered track failures in the 1800s when long lengths of track were laid without expansion joints, sliding joints or other provisions to account for the changes in length of the rails with temperature. The same occurs for sidewalks. If electrical power lines were strung tautly in summer they would rip from their connections during winter when the lines contract in the cooler temperatures.

As an example of how much thermal expansion can change the size of a structure, consider the simple case of a copper wire. Assume that a pure copper wire 30 meters long is laid out on the ground when the temperature is 30°C. What happens to the length of the length of the wire when the temperature cools to -10°C? The linear coefficient of thermal expansion of pure copper is 17 x 10^-6 meter/meter per °C. That means that for a change of 1°C a 1 meter length of copper will change 17 micrometers in length, expanding if heated and contracting if cooled. For the wire laying on the ground in this example, the length of the wire would change 0.0204 meters or just over 2 cm. That produces a strain (change in length per unit length) of 0.068%.

That may not seem like much, but what if the wire was not laying on the ground but instead had been pulled taut between two electric poles? Ignoring the force needed to pull the wire taut, how much stress would be generated from the thermal contraction of the wire? Assuming elastic behavior (no permanent deformation, similar to a rubber band), the stress is related to the strain by Young's Modulus (E). The strain times the Young's modulus gives the stress generated by that strain (or the stress needed to cause that strain). For copper, the Young's modulus is 117 gigapascal (GPa) or GPa/(m/m). Multiplying the strain (0.00068 m/m) by Young's Modulus (117 GPa) tells us that the stress generated would be 80.73 megapascal (MPa). that is about twice the stress needed to cause permanent plastic deformation of the copper and more than sufficient to pull a wire out of a simple mechanical connection or adhesive joint.

So this demonstrates in a simple one dimensional case that thermal expansion can generate large forces. What about liquid mercury?

Your question as stated requires a combination of assumptions, some realistic and some purely hypothetical. The mercury will be allowed to expand realistically, but the containing material will be assumed to have absolutely no thermal expansion occuring while the mercury is being heated. In reality the container would also expand some amount. Even in a mercury thermometer the glass containing the mercury expands some. A thin wall will be assumed to simplify the calculation of the stresses and analysis of the pressure vessel. Elastic behavior will initially be assumed and the resulting stresses compared to the yield and ultimate tensile strengths of some common metals.

This analysis is done in two steps. First the volume of the mercury is allowed to expand freely to a new volume based upon the thermal expansion of mercury as if there were no containing vessel. A hydrostatic pressure is then applied to the mercury to force it to return to its original volume. This hydrostatic pressure is equal to the pressure acting on the pressure vessel. The hydrostatic pressure will determine the stress in the walls of the pressure vessel.

The volume or cubical expansion of mercury is 0.181 x 10^-3 cm³ per cm³ per °C. That means the volume of 1 cubic centimeter of mercury will increase by 0.181 x 10^-3 cm³ when the mercury is heated 1°C.

The derivation of the relationship between the pressure inside a spherical pressure vessel and the stress in the walls is available in most civil and mechanical engineering text books such as Mechanics of Materials by Beer and Johnston (pp.325-328). The end result is that the stress in the wall is equal to the pressure times the radius divided by four times the thickness of the wall or s=pr/4t.

Hayward published the isothermal bulk modulus of mercury at 20°C and 192 kilobars or kbar (19.2 GPa) as 251.7 kbar (25.17 GPa) ±0.4%. Using additional data the modulus as a function of pressure was determined to be 250.5 + 4.5P kbars (25.05 +0.45P GPa). Since the compressibility of the mercury is equal to the pressure required to change the volume of the mercury a given amount divided by the original volume. For mercury, it requires about 250 MPa to compress 1 cubic centimeter of mercury to 0.99 cubic centimeters.

For the pressure vessel, assume a radius (r) of 1 m and a wall thickness (t) of 0.01 m. The stress will be equal to 1/(0.01*4) or 25 times the pressure. EXCEL or a calculator can now be be used to calculate the stresses using the values for the cubical thermal expansion and compressibility of mercury assuming various temperatures up to its boiling point of 357°C. Example values are given in the table below. For simplicity a constant compressibility value of 25.05 GPa was used rather than iterating the value to take into account the change in compressibility with pressure.

How do these values compare to some real life materials? The yield stress is the stress at which a material will begin to undergo plastic (permanent) deformation due to an applied stress. The ultimate tensile strength is the maximum stress that can be sustained by a material without failure. Example values are given below.

References:
Metals Handbook Desk Ed., ASM, Metals Park, OH (May 1995), pp.15-1 to 15-7)
Smithells Metals Reference Handbook, Eighth Ed., Butterworths, London, England (1983), pp. 22-1 to 22-191

While the analysis can be refined by taking into account the change in compressibility, the thermal expansion of the container and the decrease in material properties with increasing temperature, this analysis gives a good "order of magnitude" result and allows a simple comparison of the stresses generated by thermal expansion of mercury compared to the strength of some typical metals. For a 50°C increase in temperature, the stresses generated are sufficient to permanently deform most of the alloys listed. This is fairly typical of real life situations and shows why thermal expansion must be taken into account when designing structures that undergo temperature changes.

Hopefully this is enough to get you started in a further analysis.