Consider the piston-cylinder apparatus shown in Figure...

Consider the piston-cylinder apparatus shown in Figure P20.69. The bottom of the cylinder contains 2.00 $\mathrm{kg}$ of water at just under $100.0^{\circ} \mathrm{C} .$ The cylinder has a radius of $r=7.50 \mathrm{cm} .$ The piston of mass $m=3.00 \mathrm{kg}$ sits on the surface of the water. An electric heater in the cylinder base transfers energy into the water at a rate of $100 \mathrm{W} .$ Assume the cylinder is much taller than shown in the figure, so we don't need to be concerned about the piston reaching the top of the cylinder. (a) Once the water begins boiling, how fast is the piston rising? Model the steam as an ideal gas. (b) After the water has completely turned to steam and the heater continues to transfer energy to the steam at the same rate, how fast is the piston rising? (FIGURE CANNOT COPY)

Key Concepts

Piston-Cylinder System

A piston-cylinder system is a typical setup in thermodynamics where a movable piston encloses, compresses, or expands a fluid inside a cylinder. The movement of the piston directly reflects changes in pressure, temperature, and volume of the contained substance, and the system is often used to study energy transfer, work interactions, and phase changes in a controlled manner.

Phase Change (Boiling Process)

The phase change from liquid to vapor, as evidenced by boiling, involves the energy transfer necessary to overcome molecular forces without a substantial change in temperature. In this process, the added energy is used for the latent heat of vaporization, leading to rapid volume expansion, which in turn can cause mechanical movement, such as the rising of a piston in the apparatus.

Ideal Gas Model

The ideal gas model is a simplification that assumes the gas molecules do not interact with each other except through elastic collisions and that the volume of the molecules themselves is negligible compared to the container volume. When applied to steam or superheated vapor under certain conditions, it allows for the use of the ideal gas law to relate pressure, volume, and temperature, thus simplifying the analysis of the piston’s motion during expansion.

Energy Transfer and Power

Energy transfer at a constant rate, quantified as power, is central in thermodynamic processes. In the context of a piston-cylinder apparatus, a given power input, like that from an electric heater, determines how much energy is added to the system per unit time. This added energy contributes both to increasing the internal energy of the fluid (or changing its phase) and to performing work on the piston by causing it to move.

Pressure Equilibrium and Force Balances

Pressure equilibrium in a piston-cylinder apparatus is established by the balance between the pressure exerted by the fluid inside and the external forces acting on the piston, including its weight and atmospheric pressure. This force balance sets the conditions under which the piston remains in static or dynamic equilibrium, thereby linking macroscopic mechanical motion to the thermodynamic state variables of the system.

Work Done by Expanding Gas

The work done by an expanding gas is determined by the product of the pressure and the change in volume, which in the case of a moving piston can be expressed as the pressure multiplied by the piston’s displacement and the cross-sectional area of the cylinder. When energy is continuously supplied to the system, a part of it is converted into work that results in the motion of the piston, and understanding this partitioning of energy is essential for analyzing the dynamics of the process.

Transcript

00:01 Hi everyone this is the problem based on heating of water amount of the heat aftured by the water and conversion of heat into mechanical work here it is given in a piston cylinder system water is boiling having the mass 2 kg temperature of water is 100 degrees celsius radius of cylinder is 0 .075 meter mass of the piston is 3k g.

00:45 Rate of supply of heat is 100 watt.

00:54 In the first part we have to find the velocity at which the piston is rising when water is boiling and in second part we have to find the speed of water when heat continue to supply to convert stream of water.

01:55 Let us here the first.

01:56 The first of water, the first of the part density of water is sorry density of steam is 0 .59 kg.

02:23 Per meter cube and latent heat of pepperization is 2 .26 into 10 to the power 6 jule per kj.

02:37 Power given is called to latent heat of paparization into rate of conversion of water into steam mass will be density into area into velocity here dm upon d t will be row into volume volume will be area into distance so dm upon d t will be row into pi r square d s upon d t will be the speed by which cylinder is rising up so from here you can write pi r square, which speed will be p upon.

04:20 Now substitute the value, we will get the answer.

04:24 Power is given 100 watt.

04:27 Latent heat of pepperization is 2 .26.

04:30 10 to the power 6.

04:33 Density is 0 .59 into pi...

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