Final answer:
Using the ideal gas law, one can predict how temperature affects pressure in a closed system and determine the temperature at which the vessel might shatter due to increased internal pressure. However, without specific numerical data, we cannot calculate the exact temperature rise that would cause the vessel to burst.
Step-by-step explanation:
The scenario described deals with the behavior of gases under varying temperatures and the concept of pressure within a closed container, which is a fundamental topic in Physics, specifically thermodynamics. Using the ideal gas law (PV = nRT), where P is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is temperature, we can understand how temperature affects pressure in a closed system. If the initial state of a gas is known, and the volume of the container does not change, any rise in temperature will result in an increase in pressure. To find out at what temperature the vessel containing gas will shatter, one would compare the vessel's pressure limit with the pressure of the gas at the increased temperature and solve for the temperature.
In the absence of specific values for initial temperature, initial pressure, and the exact pressure at which the vessel will shatter, we are not able to calculate the exact temperature. However, an understanding of the safety factor and the critical point of substances provides insight into how materials behave under high pressure and temperature conditions beyond which they retain neither purely liquid nor purely gaseous characteristics, thus becoming supercritical fluids.