[Kr.I.S (OP)]

How will temperature effect mass in a relativistic way?
for example how would the relativistic mass of a system of particles - A, with n number of particles and temperature T where all particles have rest mass m differ to an identical system of particles - B, but with temperature T+x ?

[Kryptid]

If you're asking for an equation, I'm afraid I can't supply any directly of the sort which you are asking for. However, you might be able to approximate the mass increase of the system if you know the average velocity of the particles that make up that system.

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/releng.html

Scroll down to where it says "Relativistic Kinetic Energy" and you should see a handy calculator.

[lightarrow]

How will temperature effect mass in a relativistic way?
for example how would the relativistic mass of a system of particles - A, with n number of particles and temperature T where all particles have rest mass m differ to an identical system of particles - B, but with temperature T+x ?

It's even rest mass which differ, not only relativistic mass. About the relation between mass/energy and temperature, it depends on the specific heat, which depends on how you heat the object, on which material is made of, and on temperature itself. Very rougly, the average energy of a single particle of the object is kT so knowing T and the object's number of particles you can have some idea of the energy you have added to it starting from (say) room temperature, that is its increase in (rest) mass.

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[Pmb]

How will temperature effect mass in a relativistic way?
for example how would the relativistic mass of a system of particles - A, with n number of particles and temperature T where all particles have rest mass m differ to an identical system of particles - B, but with temperature T+x ?

An increase in temperature of a body is a result of an input of thermal energy. This increase in energy causes an increase in mass according to the expression E = mc². The relationship between temperature and thermal energy depends on the material.

The temperature of a body is a reflection of the average kinetic energy of the particles that make up the body. I created a web page which illustrates how an increase in the kinetic energy of the particles of a body contributes to the mass of the body. See http://home.comcast.net/~peter.m.brown/gr/weight_moving_body.htm

As you can easily see a body at rest can be rotating in its rest frame. The rest frame is defined as the frame of reference in which the total momentum of the body is zero. I illustrated this in another web page I created. To see how the rotational kinetic energy of a body causes an increase in rest mass of the body please see my website at http://home.comcast.net/~peter.m.brown/sr/rotating_cylinder.htm