question: If heat is a measure of the mean kinetic energy of the particles of a substance, then how does compressing it increase its temperature, since compression squishes the particles of a substance closer together, theoretically limiting movement?
comment: You're thinking temperature, not heat. These are related but different. The reason compression increases temperature is because the energy of the compressed gas is now more concentrated into a smaller volume, and thus the mean kinetic energy of the *volume* of the substance is higher because it occupies less volume with the same stuff.
passages: However, this computation is valid only all parts of the object are at the same external pressure before and after the measurement. That may not be possible in some cases. For example, when heating an amount of gas in an elastic container, its volume "and pressure" will both increase, even if the atmospheric pressure outside the container is kept constant. Therefore, the effective heat capacity of the gas, in that situation, will have a value intermediate between its isobaric and isochoric capacities formula_8 and formula_9.
   In transition regions, where this pressure dependent dissociation is incomplete, both beta (the volume/pressure differential ratio) and the differential, constant pressure heat capacity greatly increases.
   so our adiabatic constant for this example is about 6.31 Pa m.
  
  The gas is now compressed to a 0.1 L (0.0001 m) volume (we will assume this happens quickly enough that no heat can enter or leave the gas through the walls). The adiabatic constant remains the same, but with the resulting pressure unknown
  
  so solving for "P":
   In order to read a compressibility chart, the reduced pressure and temperature must be known. If either the reduced pressure or temperature is unknown, the reduced specific volume must be found. Unlike the reduced pressure and temperature, the reduced specific volume is not found by using the critical volume. The reduced specific volume is defined by,
  
  where formula_36 is the specific volume.
   Once two of the three reduced properties are found, the compressibility chart can be used. In a compressibility chart, reduced pressure is on the x-axis and Z is on the y-axis. When given the reduced pressure and temperature, find the given pressure on the x-axis. From there, move up on the chart until the given reduced temperature is found. Z is found by looking where those two points intersect. the same process can be followed if reduced specific volume is given with either reduced pressure or temperature.
  
  Section::::Observations made from a generalized compressibility chart.
   A power "P" is applied e.g. as electrical power. If the compression is adiabatic, the gas temperature goes up. In the reversible case it would be at constant entropy, which corresponds with a vertical line in the "T"–"s" diagram. For example, compressing nitrogen from 1 bar (point a) to 2 bar (point b) would result in a temperature increase from 300 K to 380 K. In order to let the compressed gas exit at ambient temperature "T", heat exchange, e.g. by cooling water, is necessary. In the ideal case the compression is isothermal. The average heat flow to the surroundings is "Q̇". Since the system is in the steady state the first law gives
   As temperature and pressure increase along the coexistence curve, the gas becomes more like a liquid and the liquid becomes more like a gas. At the critical point, the two are the same. So for temperatures above the critical temperature (126.2 K), there is no phase transition; as pressure increases the gas gradually transforms into something more like a liquid. Just above the critical point there is a range of pressure for which "Z" drops quite rapidly (see the 130 K curve), but at higher temperatures the process is entirely gradual.
   BULLET::::- If the pressure is kept constant (for instance, at the ambient atmospheric pressure), and the sample is allowed to expand, the expansion generates work as the force from the pressure displaces the enclosure or the surrounding fluid. That work must come from the heat energy provided. The specific heat thus obtained is said to be measured at constant pressure (or isobaric), and is often denoted formula_20, formula_21, formula_22, etc.
   where ρ is the density of the substance under the applicable conditions.
  
  The corresponding expression for the ratio of specific heat capacities remains the same since the thermodynamic system size-dependent quantities, whether on a per mass or per mole basis, cancel out in the ratio because specific heat capacities are intensive properties. Thus: 
  
  The difference relation allows one to obtain the heat capacity for solids at constant volume which is not readily measured in terms of quantities that are more easily measured. The ratio relation allows one to express the isentropic compressibility in terms of the heat capacity ratio.
   where "p" is the pressure of the gas, "T" is its temperature, and formula_9 is its molar volume. In the case of an ideal gas, the compressibility factor "Z" is equal to unity, and the familiar ideal gas law is recovered:
  
  "Z" can, in general, be either greater or less than unity for a real gas.
   Here, work is entirely consumed by expansion against the surroundings. Of the total heat applied (709.3 kJ), the work performed (202.7 kJ) is about 28.6% of the supplied heat.
   In uniaxial compression, the forces are directed along one direction only, so that they act towards decreasing the object's length along that direction. The compressive forces may also be applied in multiple directions; for example inwards along the edges of a plate or all over the side surface of a cylinder, so as to reduce its area (biaxial compression), or inwards over the entire surface of a body, so as to reduce its volume.
   BULLET::::- On the other hand, if the expansion is prevented — for example by a sufficiently rigid enclosure, or by increasing the external pressure to counteract the internal one — no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the object, including raising its temperature by an extra amount. The value obtained this way is said to be the molar heat capacity at constant volume (or isochoric) and denoted "c", "c", "c", etc.
   As for the compressibility of gases, the principle of corresponding states indicates that any pure gas at the same reduced temperature, formula_18, and reduced pressure, formula_19, should have the same compressibility factor.
  
  The reduced temperature and pressure are defined by
   BULLET::::- If the pressure is kept constant (for instance, at the ambient atmospheric pressure), and the sample is allowed to expand, the expansion generates work as the force from the pressure displaces the enclosure. That work must come from the heat energy provided. The value thus obtained is said to be the molar heat capacity at constant pressure (or isobaric), and is often denoted "c", "c", "c", etc.
   Applying the ideal gas law, this becomes
  
  assuming that the quantity of gas stays constant, e.g., there is no phase transition during a chemical reaction. According to the equipartition theorem, the change in internal energy is related to the temperature of the system by
  
  where "c" is specific heat at a constant volume.
  
  Substituting the last two equations into the first equation produces:
  
  where "c" is specific heat at a constant pressure, and the "c+R" was brought to "c" by Mayer's relation.
  
  Section::::Specific heat capacity.
   Since enthalpy and internal energy are independent of pressure,
  
  As in the first example, about 28.6% of the supplied heat is converted to work. But here, work is applied in two different ways: partly by expanding the surrounding atmosphere and partly by lifting 10,332.2 kg a distance "h" of 1 m.
  
  Thus half the work lifts the piston mass (work of gravity, or “useable” work), while the other half expands the surroundings.
   An adiabatic process is one where there is no heat transfer between the fluid and the surroundings: the system is insulated against heat transfer. If the process is furthermore internally reversible (smooth, slow and frictionless, to the ideal limit) then it will additionally be isentropic.
   An isobaric process is shown on a "P"–"V" diagram as a straight horizontal line, connecting the initial and final thermostatic states. If the process moves towards the right, then it is an expansion. If the process moves towards the left, then it is a compression.
  
  Section::::Sign convention for work.
  
  The motivation for the specific sign conventions of thermodynamics comes from early development of heat engines. When designing a heat engine, the goal is to have the system produce and deliver work output. The source of energy in a heat engine, is a heat input.
   BULLET::::- On the other hand, if the expansion is prevented — for example by a sufficiently rigid enclosure, or by increasing the external pressure to counteract the internal one — no work is generated, and the heat energy that would have gone into it must instead contribute to the internal energy of the sample, including raising its temperature by an extra amount. The specific heat obtained this way is said to be measured at constant volume (or isochoric) and denoted formula_23, formula_24 formula_25, etc.
   As seen above, the behavior of "Z" is qualitatively similar for all gases. Molecular nitrogen, N, is used here to further describe and understand that behavior. All data used in this section were obtained from the NIST Chemistry WebBook. It is useful to note that for N the normal boiling point of the liquid is 77.4 K and the critical point is at 126.2 K and 34.0 bar.
   Section::::Derivation and use.:Landau's explanation.
  
  According to Landau, "In compression or expansion, as in any rapid change of state, the fluid ceases to be in thermodynamic equilibrium, and internal processes are set up in it which tend to restore this equilibrium. These processes are usually so rapid(i.e. their relaxation time is so short) that the restoration of equilibrium follows the change in volume almost immediately unless, of course, the rate of change of volume is very large."
   This can be seen in the graph showing the high temperature behavior. As temperature increases, the initial slope becomes less negative, the pressure at which "Z" is a minimum gets smaller, and the pressure at which repulsive interactions start to dominate, i.e. where "Z" goes from less than unity to greater than unity, gets smaller. At the Boyle temperature (327 K for N), the attractive and repulsive effects cancel each other at low pressure. Then "Z" remains at the ideal gas value of unity up to pressures of several tens of bar. Above the Boyle temperature, the compressibility factor is always greater than unity and increases slowly but steadily as pressure increases.
   The easiest way to get an equation of state linking "P" and "V" is to assume that "K" is constant, that is to say, independent of pressure and deformation of the solid, then we simply find the Hooke's law. In this case, the volume decreases exponentially with pressure. This is not a satisfactory result because it is experimentally established that as a solid is compressed, it becomes more difficult to compress. To go further, we must take into account the variations of the elastic properties of the solid with compression.
   Starting with the flow pattern observed in both theory and experiments, the increased flow speed over the upper surface can be explained in terms of streamtube pinching and conservation of mass.
  
  For incompressible flow, the rate of volume flow (e.g. volume units per minute) must be constant within each streamtube since matter is not created or destroyed. If a streamtube becomes narrower, the flow speed must increase in the narrower region to maintain the constant flow rate, to satisfy the principle of conservation of mass.
  
presuppositions: If heat is calculated by the measure of kinetic energy of particles within a substance, than compressing said particles should not increase heat. Temperature and heat are the same. 
corrections: Because the kinetic energy has been compressed, the kinetic energy actually increases because it is moving within an area with much less volume. Temperature and heat are related, but temperature and heat are different. 
raw_presuppositions: If heat is calculated by the measure of kinetic energy of particles within a substance, than compressing said particles should not increase heat. Temperature and heat are the same.  movement of particles is limited by compressing them
raw_corrections: Because the kinetic energy has been compressed, the kinetic energy actually increases because it is moving within an area with much less volume. Temperature and heat are related, but temperature and heat are different.  The concentration of particles is greater in a condensed space, which increases kinetic energy and temperature.