SAT Physics Thermal Properties - Thermal Systems

SAT Physics Thermal Properties - Thermal Systems

THERMAL SYSTEMS
In mechanics, the focus was often on the motion of a single object, such as a block sliding down an incline. However, when several objects experience the same motion, such as two blocks tied together by a string, then they can be treated as a single system. This allows the problem to be solved as though the blocks and the string are one larger object. Treating several objects experiencing the same conditions as one aggregate object is extremely useful in thermal physics.
    The objects involved in thermal physics problems are the countless atoms and molecules that make up substances. Working with single atoms is not possible. As a result, quantities and equations that describe the particles acting together as a single system have been established. The system might be an ice cube that melts into a liquid and vaporizes into a gas. It could be an iron rod that is heated and expands. It could even be a gas trapped in a cylinder with a movable piston. Some aspects of thermal physics involve analyzing the individual objects (atoms) within the larger system. Other problems will be concerned with the properties, trends, and quantities that describe the system as a whole.

Table 18.1. Variables That Describe Thermal Properties
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THERMAL ENERGY
The atoms comprising solids, liquids, and gases are in constant motion. Thus, each atom has a tiny amount of kinetic energy. The chemical bonds and intermolecular forces holding substances together store tiny amounts of potential energy. Thermal energy is the sum of the microscopic kinetic and potential energies of all the atoms comprising the system under investigation.
    Thermal energy increases when substances become hotter. When solids become hotter, their molecules vibrate faster and increase in kinetic energy. When liquids and gases become hotter, the randomly moving molecules move faster and increase in kinetic energy.

Converting Mechanical Energy into Thermal Energy
Thermal energy and mechanical energy are both measured in joules. According to the law of conservation of energy, these two forms of energy are capable of transforming from one type into the other as long as the total amount of energy in the system is conserved. This chapter will explore instances where mechanical energy is transformed into thermal energy. This transfer occurs in two common ways: friction and inelastic collisions.
    When a block slides down a rough incline, the rubbing of the block against the incline makes the molecules in both the block and the incline vibrate faster. Their thermal energies increase. Conservation of energy dictates that the thermal energy gained must equal the kinetic energy lost by the slowing block. The kinetic energy lost by the block is an energy change, and an energy change is known as work. In this case, the work of friction transfers kinetic energy from the block into thermal energy. The work of friction is equal to the kinetic energy lost by the block and is also equal to the thermal energy generated.
    In inelastic collisions, the colliding objects impact one another. The impact causes the atoms within the objects to vibrate faster. In an inelastic collision, kinetic energy is lost. As with friction, the thermal energy generated in the inelastic collision is equal to the kinetic energy lost during an inelastic collision.

TEMPERATURE
Temperature is a relative measure of hot and cold. The Celsius and Fahrenheit temperature scales were established to encompass common extremes of hot and cold experienced by humans. The extremes of freezing and boiling water were used to set the 0° and 100° marks on the Celsius scale. Once established and accepted, the Celsius temperature scale became a means to quantify temperatures so that hot and cold objects could be numerically compared with one another.
Temperature reflects the speed and microscopic kinetic energy of the molecules of a substance. This means that the temperature of an object reflects the thermal energy of the object. Raising the temperature of an object increases its hotness. This causes the atoms of the substance to move faster, increasing the thermal energy. In order to relate temperature directly to thermal energy mathematically, a less arbitrary temperature scale is needed. Experimentation with gases resulted in projecting a temperature at which all molecular motion should cease. At this temperature, gas molecules would have no microscopic kinetic energy and therefore no thermal energy. The resulting temperature scale is the Kelvin temperature scale. Absolute zero, 0 kelvin, on this scale is equal to -273°C. It is the point where both temperature and thermal energy equal zero. As a result, formulas could be developed that use the Kelvin temperature to calculate thermal energy.
    Which scale should be used for calculations? If a formula contains a T for temperature, then the Kelvin temperature must be used. However, if a formula contains a AT, then either the Celsius or Kelvin scale can be used. Why? Although the Celsius and Kelvin temperature scales have different zero points, 1 degree Celsius is equal to 1 kelvin. Therefore, a change in temperature, AT, will be the same using either scale. When in doubt, it is always safer to use the Kelvin scale. To convert from degrees Celsius to kelvin, add 273.

THERMAL EXPANSION
When a solid or a liquid is heated, the atoms and molecules vibrate faster, causing a substance that is heated to expand. This process is known as thermal expansion. Conversely, a substance that is cooled will contract. Heating and cooling will change the length of linear objects, the surface area of two-dimensional objects, and the volume of three-dimensional objects.
    In conceptual problems any type of object can be given and the effects of heating or cooling it may be asked. Essentially the entire object expands or contracts proportionally.

Linear Expansion
When a linear object, such as a metal rod, is heated, its length will increase, as shown in Figure 18.1.
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Figure 18.1. Linear expansion
 The amount that the rod increases in length, AL, is calculated as follows:
ΔL = αL0ΔT
ΔL = αL0( Tf - Ti )
   The change in length, ΔL, is equal to the product of the coefficient of linear expansion, α, the original length, L0, and the change in temperature, ΔT. The coefficient of linear expansion is a physical property of the substance that is expanding. Its value depends on the composition of the object. When a substance is heated, its final temperature, Tf will be larger than its initial temperature, Ti. This heating will cause ΔL to become positive, indicating that length is increasing. However, when a substance is cooled, Ti will be larger than Tf This cooling will result in a negative ΔL, indicating that length is decreasing. To find the new length, L, simply add the change in length, ΔL, to the starting length, L0.
    The linear expansion formula can be used to solve for the expansion of a rectangular area. A rectangle has a length and a width. Simply solve for linear expansion twice, once for each dimension. Then multiply the new length by the new width to find the new area.

Linear Expansion
A 5.0-meter-long iron rod is heated from 0°C to 100°C. The linear coefficient of expansion for iron is 12 x 10"6 K"1. Determine the change in length of the rod.
 
WHAT'S THE TRICK?

Use the linear expansion formula, but read the question carefully. Is the question looking for the change in length or the new length after expanding or contracting? This question simply requests the change in length.
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IDEAL GASES
Gases are extremely chaotic. They consist of countless particles moving at different speeds in every possible direction. These particles collide with each other and with surfaces. Analyzing the behavior of a gas by examining individual gas molecules is impossible. Instead, a gas is treated as a single aggregate system. Due to its complexity, a simplified model of a gas system was developed. This model is known as the ideal gas model. Certain assumptions are made about ideal gases:
  • The particles are so small that the volume of the particles is negligible.
  • The attraction between particles is zero (no microscopic potential energy).
  • The particles are in constant random motion (microscopic kinetic energy).
  • Collisions between particles and with surfaces are perfectly elastic.
These assumptions make it possible to analyze ideal gases using the mathematical relationships discussed in the sections that follow. The behavior of a real gas at normal temperatures is nearly identical to that of an ideal gas.
    Since the particles of a gas move at varying speeds in all directions, calculations involving specific directional velocities are impossible. As a result, only the average speed of the gas particles can be determined. As temperature increases, average speed increases. The average speed, vrms, is related to the Kelvin temperature, T, of the gas according to the following formula:
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   This formula is unlikely to be used on the exam because it requires the use of a calculator. However, you should know that the speed of a gas is changed by the square root of the factor applied to the Kelvin temperature. If the Kelvin temperature is doubled, the average speed of the gas particles increases by V2. The variable R is the universal gas constant. In physics, it has the value of 8.31 joules per moles • kelvin. The variable M is the molar mass of the gas particles in kilograms per mole.
    The energy of a gas is related to the average speed and kinetic energy of the gas particles. Since the attraction between the particles of an ideal gas is assumed to be zero, the only type of energy that gas particles possess is the microscopic kinetic energy due mainly to their speed. Since all the particles have different speeds, the kinetic energy, K, of the gas particles is also an average. It is calculated as follows:
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   The variable kB is Boltzmann’s constant, kB = 1.38 x 10~23 joules/kelvin. The nature of this value makes this formula unlikely to appear on an exam that does not allow you to use a calculator. However, the formula is valuable in demonstrating that the average kinetic energy of a gas is directly proportional to its Kelvin temperature. If the Kelvin temperature of a gas is doubled, the average kinetic energy of the gas particles also doubles. Be careful. This is not true for temperatures involving the Celsius scale. Doubling the Celsius temperature does not double the Kelvin temperature. Again when in doubt, use the Kelvin temperature scale.
    The trends relating temperature, gas particle speed, and the energy of the gas may be the most important information to retain. As temperature increases, molecules move faster and their kinetic energy increases. If the attraction between the gas particles is assumed to be zero, the microscopic potential energy between gas particles is also zero. This means that the thermal energy of a gas is essentially the sum of the kinetic energies of all the gas particles.

Pressure
When a particle of gas collides with a surface, such as the walls of a container, the momentum change (impulse) experienced by the gas particle causes a force to be applied to the surface. Calculating the force of each particle of gas and adding them up would be an impossible task. However, the aggregate force of all gas particles treated as a single system striking a specific area can be measured and is a valuable quantity. The force, F, due to all the collisions of gas particles striking 1 square meter of surface area, A, is known as pressure, P.
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   Pressure is measured in pascals (Pa). A pascal is equal to a newton per meter squared (N / m2).
Once again, the overall trends are extremely important. As the temperature of a gas increases, the particles of gas move faster and increase their speed. The kinetic energy of the gas particles increases as does the thermal energy of the gas as a whole. In addition, the faster and more energetic gas particles experience greater momentum changes when they strike surfaces, resulting in an increase in the pressure of the gas.

Ideal Gas Law
The ideal gas law is a mathematical relationship relating the pressure P, volume V, number of moles n, and temperature Tof an ideal gas.
PV = nRT
   Volume in physics is measured in meters cubed, m3. The variable n represents the number of moles of gas particles. For students who have not yet completed chemistry, a mole is similar to terms such as “pair” or “dozen.” These terms bring specific numbers to mind (pair = 2 and dozen = 12). A mole is a specific number of particles. One mole is 6.02 x 1023 particles, and this value is known as Avogadro’s number. Avogadro’s number is not needed for the ideal gas law; just the number of moles is needed. If a problem specifies one mole (1 mol) of gas, then n = 1. For two moles of gas, n = 2, etc. The ideal gas constant, R, is equal to 8.31 joules per mol • kelvin.
    Conceptual problems will focus on relationships between key variables in the ideal gas law: PV = nRT.
  1. Pressure and volume are inversely proportional to each other. For example, if the temperature and the moles of gas are held constant, then a decrease in volume is compensated by an increase in pressure.
  2. Pressure is directly proportional to the Kelvin temperature of a gas. For example, increasing the temperature while holding the volume and the number of moles constant will increase the pressure of a gas.
  3. Volume is directly proportional to the Kelvin temperature of a gas. For example, increasing the temperature while holding the pressure and the number of moles constant will increase the volume of a gas.

Ideal Gas Law
A gas is trapped in a cylinder with a movable piston. How is the pressure of the gas affected if the temperature of the gas doubles while the piston moves inward, reducing the volume by half?
 
WHAT'S THE TRICK?

The ideal gas law is the key. The problem states that the gas is trapped, implying that the number of moles remains constant. Any value remaining constant, such as the number of moles and the gas constant, cannot cause change. Ignore them.
PV = nRT
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The pressure must quadruple in order to maintain the equality.

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