Rituraj Tiwari
Last Activity: 4 Years ago
To determine the pressure exerted by the liquid at the bottom of a thin-walled cylindrical metal vessel when both the vessel and its contents are heated, we need to consider the effects of thermal expansion on both the vessel and the benzene contained within it. Let's break this down step by step.
Understanding Thermal Expansion
When materials are heated, they expand. This phenomenon is quantified by coefficients of linear and volume expansion. The linear coefficient of expansion for the vessel is given as 10-3 °C-1, while the volume expansion coefficient for benzene is also 10-3 °C-1. This means that for every degree Celsius increase in temperature, the dimensions of the vessel and the volume of benzene will change according to these coefficients.
Calculating Changes in Dimensions
- Vessel Expansion: The length change (ΔL) of the cylindrical vessel can be calculated as:
ΔL = L₀ * α * ΔT
where L₀ is the original length, α is the linear coefficient of expansion, and ΔT is the change in temperature. Since we are looking at a cylindrical shape, the area can also change, affecting the pressure exerted by the liquid. - Benzene Expansion: For the benzene, the volume change (ΔV) can be expressed as:
ΔV = V₀ * β * ΔT
where V₀ is the original volume, β is the volume expansion coefficient, and ΔT is the same temperature change.
Pressure Calculation
The pressure due to the benzene at the bottom of the vessel can be calculated using the hydrostatic pressure formula:
P = ρ * g * h
where:
- P = Pressure at the bottom
- ρ = Density of the liquid (benzene)
- g = Acceleration due to gravity (approximately 9.81 m/s²)
- h = Height of the liquid column
As the benzene expands, its volume increases, which may also affect the height of the liquid column, increasing the pressure at the bottom. If the volume of the benzene increases significantly due to heating, the height (h) can be recalculated based on the new volume divided by the cross-sectional area of the cylinder.
Final Considerations
In summary, when both the vessel and the benzene are heated by 10°C, the pressure at the bottom of the vessel will increase due to the expansion of the liquid. The exact numerical value of the pressure would depend on the initial conditions such as the original volume of benzene, the cross-sectional area of the cylinder, and the density of the benzene at the initial temperature. To find the precise pressure, you would need to plug the specific values into the equations we discussed.
This understanding of thermal expansion and pressure dynamics is crucial in many engineering applications, from designing safe containers for liquids to ensuring the stability of structures subjected to temperature changes.
