Question

first law of thermodynamics
thermodynamic sign conventions for heat and work
process
convention
heat added to system
$q > 0$
heat removed from system
$q < 0$
work done by system
$w > 0$
work done on system
$w < 0$
gas
$v, p, t$
state
variables
$de_{12}=e_f - e_i = e_{in}(2)-e_{in}(1)$
$q, w, de$
$q = de+w$
$1$st law.
$e_{in}=\frac{f}{2}nrt$
thermal
energy
$f$ degrees of freedom
$f = 3$: translation
diatomic
$2$: rotation
$2$: vibration

Explanation:

Step1: Identify the first - law formula

The first law of thermodynamics is given by $Q=\Delta E + W$, where $Q$ is heat, $\Delta E$ is the change in internal energy of the system, and $W$ is work done.

Step2: Understand the sign conventions

Heat added to the system ($Q>0$), heat removed from the system ($Q < 0$), work done by the system ($W>0$), and work done on the system ($W < 0$).

Step3: Relate to internal - energy formula

The internal energy of an ideal gas is $E_{in}=\frac{f}{2}nRT$, where $f$ is the degrees of freedom, $n$ is the number of moles, $R$ is the gas constant, and $T$ is the temperature. For a monatomic gas, $f = 3$ (translational degrees of freedom), for a diatomic gas, $f=5$ (3 translational + 2 rotational at low - temperatures, and may include 2 vibrational at high - temperatures).

Answer:

The first law of thermodynamics $Q=\Delta E+W$ along with sign conventions for heat and work and the internal - energy formula $E_{in}=\frac{f}{2}nRT$ are key concepts in thermodynamics. The degrees of freedom $f$ determine the internal energy of the gas, with different values for different types of gases (monatomic, diatomic etc.).