Reduce the volume of a gas.How much heat, how much pressure?

[peanutaxis]

If I, say, half the volume of a[n ideal] gas adiabatically, how on Earth can I tell how much the pressure will increase and how much the temperature will increase? PV=nRT

Also, by what mechanism does the temperature increase? What would convince the molecules to move any faster? I can see why the pressure would increase due to a larger number of collisions per unit area per unit time, but why would the average speed of a molecule increase?

Last question. If I compress a gas very quickly and it gets hot, and then let it cool off to ambient, will the pressure climb rapidly as I compress it and then drop a little as it cools due to PV=nRT?thanks!
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[Andrew Mason]

If I, say, half the volume of a[n ideal] gas adiabatically, how on Earth can I tell how much the pressure will increase and how much the temperature will increase? PV=nRT

If it is compressed adiabatically, one has to use the adiabatic condition: $PV^\gamma = K$, K being a constant and $\gamma = \frac{C_p}{C_v}$, i.e. the ratio of specific heats at constant pressure and constant volume. This assumes a reversible compression. In the real world adiabatic compressions are very close to reversible if T and P are fairly uniform throughout the volume of gas during the compression process.

Also, by what mechanism does the temperature increase? What would convince the molecules to move any faster? I can see why the pressure would increase due to a larger number of collisions per unit area per unit time, but why would the average speed of a molecule increase?

Since you are doing work on the gas with no heat flow into or out of the gas, the first law tells you that internal energy has to increase. As the walls move in, the moving walls add energy to the molecules that collide with it.

Last question. If I compress a gas very quickly and it gets hot, and then let it cool off to ambient, will the pressure climb rapidly as I compress it and then drop a little as it cools due to PV=nRT?

Yes.

AM