Each of the following nuclides is known to undergo radioactive decay by production of a beta particle, ^0_-1e. Write a balanced nuclear equation for each process. (Use 'e' for an electron, 'p' for a positron, and 'n' for a neutron. Omit states-of-matter from your answer.)

a.) ^14_6 C
b.) ^140_55 Cs
c.) ^234_90 Th

You make the subscripts add up on both sides and the mass number add up on both sides as in the first one which I will do for you completely.

614C ==> X + -10

The first step shows the emission of a beta particle (an electron).
Step 2 is to make the subscripts and superscripts add up so the left side and the right side have equal numbers. Since the electron in -1 (subscript), then X must have a subscript of 7 (7-1=6). The superscript must be 14 (14+0=14), Then you look on the periodic table and find the element (X) with an atomic number of 7. That is N. The final equation, step 3, looks like this.
614C ==> 714N + -10e

I made a typo on the first one. I omitted e from -1 ^0e.

How did you get N?

It's in my first post. I've bold faced it for you.

The first step shows the emission of a beta particle (an electron).
Step 2 is to make the subscripts and superscripts add up so the left side and the right side have equal numbers. Since the electron in -1 (subscript), then X must have a subscript of 7 (7-1=6). The superscript must be 14 (14+0=14), Then you look on the periodic table and find the element (X) with an atomic number of 7. That is N.

To write a balanced nuclear equation for each process, we need to understand how radioactive decay by beta particle emission occurs. Beta decay involves the transformation of a neutron in the nucleus into a proton, accompanied by the emission of a beta particle. A beta particle can be an electron or a positron, which is represented as ^0_-1e for an electron and ^0_+1e for a positron.

Let's write the balanced nuclear equations for each nuclide:

a.) ^14_6C undergoes beta decay:
^14_6C → ^14_7N + ^0_-1e

In the equation above, a neutron (^1_0n) in the carbon-14 nucleus undergoes beta decay and becomes a proton (^1_1p), forming nitrogen-14 (^14_7N) and emitting a beta particle (^0_-1e).

b.) ^140_55Cs undergoes beta decay:
^140_55Cs → ^140_56Ba + ^0_-1e

In this case, a neutron (^1_0n) in the cesium-140 nucleus undergoes beta decay and becomes a proton (^1_1p), forming barium-140 (^140_56Ba) and emitting a beta particle (^0_-1e).

c.) ^234_90Th undergoes beta decay:
^234_90Th → ^234_91Pa + ^0_-1e

Here, a neutron (^1_0n) in the thorium-234 nucleus undergoes beta decay and becomes a proton (^1_1p), forming protactinium-234 (^234_91Pa) and emitting a beta particle (^0_-1e).

In summary, the balanced nuclear equations for the radioactive decay by beta particle emission for each of the given nuclides are:

a.) ^14_6C → ^14_7N + ^0_-1e
b.) ^140_55Cs → ^140_56Ba + ^0_-1e
c.) ^234_90Th → ^234_91Pa + ^0_-1e