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I am a little bit confused about what an isobar is. Its online definition is that it's an element with the same number of neutrons but a different number of protons from an element $ce{X}$.

1. To me, it doesn't make sense from the get-go, because once you change the number of protons the element changes as well so why exactly is it defined as the same element $ce{X}$ with the same number of neutrons and a different number of protons.




2. Definition of an isotope: An isotope is an element $ce{X}$ with the same number of protons and a different number of neutrons.

So to the actual question now. Isn't an isobar just an isotope? Here is an example to clarify what I mean.
If we take for example carbon $ce{^12C(p:6, n:6)}$ and turn it into an isotope it will be $ce{^13C(p:6, n:7)}$, and that makes sense, but if we turn it into an isobar it would be $ce{^13C(p:7, n:6)}$, which doesn't make sense, because it looks exactly like an isotope of nitrogen $ce{^13N(p:7, n:6)}$.

If the atomic number changes than the element changes as well. So isn't an isobar just an isotope of the following element with a smaller neutron number?

Not quite, an isotope has same number of protons ($ A- N = Z = mathrm{constant}$), but a different number of neutrons ($mathrm N$ varies; e.g. $ce{^3_color{red}{1}H}$ and $ce{^2_color{red}{1}H}$, or $ce{^235_color{red}{92}U}$ and $ce{^238_color{red}{92}U}$ are isotopes).

An isobar has a fixed number of total nucleons ($Z + N = A = mathrm{constant}$; e.g. $ce{^color{red}{40}_19K}$ and $ce{^color{red}{40}_20Ca}$, or $ce{^color{red}{3}_2He}$ and $ce{^color{red}{3}_1H}$ are isobars). Not nearly as mainstream as isotopes, but isobars are important to consider when doing mass spectroscopy.

Extra fact: For nuclei of the same number of neutrons ($A - Z = N = mathrm{constant}$), the term is isotones.

I believe the definition you found may have been a little bit misleading. Here is another definition of isobar I found: each of two or more isotopes of different elements, with the same atomic weight. An isobar is referring to completely different elements. The prefix iso- means only one component must be the same between the different elements, and in the case of an isobar: mass.

I saw in your question the example that an isobar of carbon-13 would look like so $ce{^13C(p:7, n:6)}$; however, this is not proper notation, as an isobar cannot be of the same element. Once the number of protons changes, your element is no longer the same.

Isobars are simply two different elements with the same mass while isotopes are two of the same elements with different masses.

Isobar is of more interest to physics than chemistry.

As others have explained, your definition is confusing. The one here may be clearer: isobar (Wikipedia).

In chemistry, the number of protons is most significant since it determines the number of electrons and hence the chemical behaviour. The number of neutrons is relatively unimportant: variants may be useful for labeling and may have slightly different behaviour (most noticeable for $ce{^1H}$ and $ce{^2H}$). So, isotope is used fairly frequently to discuss these variants of the elements. Isobars may have very different chemical behaviour and are unlikely to be an interesting grouping.

In nuclear physics, the number of neutrons and protons have a similar significance hence the term nuclide (Wikipedia) for atoms with a specific number of protons and neutrons is useful. Isobars have a closer connection with each other than isotopes since isobars can interconvert via beta processes relatively easily. It is rare for an atom to decay into an isotope. For example, $ce{^{14}C}$ decays into $ce{^{14}N}$ rather than $ce{^{13}C}$ .

Isobars will be relevant in some specific situations in chemistry such mass spectroscopy. So, it is still useful to understand the concept.

In a nuclear physics context, one considers neutrons and protons as two different states of one object called a nucleon. This means that nuclei which are isobars are treated as a nucleus with some number, A of nucleons but having different charges (given by number of neutrons - number of protons, since N-P differs for different nuclei with N+P=A). This is useful for studying nuclear structure, since the assumption about the strong nuclear force binding neutrons and protons is that it does not depend on the electric charge. The coulomb energy is a separate contribution.

That implies that the mass difference for a given set of isobars, say, for A=53, 53Co, 53Fe, 53,Ni, 53Mn, should depend only on the difference in the coulomb energy since the strong force doesn't distinguish between neutrons and protons. It can be shown that this mass difference should fit a parabola. If you have a multiplet with more than three members, then you can test the assumption about the charge independence of the nuclear force and perhaps learn additional information about nuclear structure just by measuring the masses of those isobars. For more detail, you can look up Isobaric Mass Multiplet Equation and/or nuclear isospin.